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Theorem sticksstones19 40573
Description: Extend sticks and stones to finite sets, bijective builder. (Contributed by metakunt, 23-Oct-2024.)
Hypotheses
Ref Expression
sticksstones19.1 (𝜑𝑁 ∈ ℕ0)
sticksstones19.2 (𝜑𝐾 ∈ ℕ0)
sticksstones19.3 𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔𝑖) = 𝑁)}
sticksstones19.4 𝐵 = { ∣ (:𝑆⟶ℕ0 ∧ Σ𝑖𝑆 (𝑖) = 𝑁)}
sticksstones19.5 (𝜑𝑍:(1...𝐾)–1-1-onto𝑆)
sticksstones19.6 𝐹 = (𝑎𝐴 ↦ (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))))
sticksstones19.7 𝐺 = (𝑏𝐵 ↦ (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍𝑦))))
Assertion
Ref Expression
sticksstones19 (𝜑𝐹:𝐴1-1-onto𝐵)
Distinct variable groups:   𝐴,𝑎,𝑖,𝑥,𝑦   𝐴,𝑏,𝑖,𝑥,𝑦   𝐵,𝑎,𝑖,𝑥,𝑦   𝐵,𝑏   𝐹,𝑏,𝑦   𝐺,𝑎,𝑥   𝐾,𝑎,𝑔,𝑖,𝑦   𝐾,𝑏,𝑔   𝑥,𝐾   𝑔,𝑁   ,𝑁   𝑆,𝑎,,𝑖,𝑥   𝑆,𝑏,   𝑦,𝑆   𝑍,𝑎,𝑔,𝑖,𝑦   𝑍,𝑏,,𝑥   𝜑,𝑎,𝑖,𝑥,𝑦   𝜑,𝑏
Allowed substitution hints:   𝜑(𝑔,)   𝐴(𝑔,)   𝐵(𝑔,)   𝑆(𝑔)   𝐹(𝑥,𝑔,,𝑖,𝑎)   𝐺(𝑦,𝑔,,𝑖,𝑏)   𝐾()   𝑁(𝑥,𝑦,𝑖,𝑎,𝑏)

Proof of Theorem sticksstones19
Dummy variables 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sticksstones19.1 . . 3 (𝜑𝑁 ∈ ℕ0)
2 sticksstones19.2 . . 3 (𝜑𝐾 ∈ ℕ0)
3 sticksstones19.3 . . 3 𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔𝑖) = 𝑁)}
4 sticksstones19.4 . . 3 𝐵 = { ∣ (:𝑆⟶ℕ0 ∧ Σ𝑖𝑆 (𝑖) = 𝑁)}
5 sticksstones19.5 . . 3 (𝜑𝑍:(1...𝐾)–1-1-onto𝑆)
6 sticksstones19.6 . . 3 𝐹 = (𝑎𝐴 ↦ (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))))
71, 2, 3, 4, 5, 6sticksstones18 40572 . 2 (𝜑𝐹:𝐴𝐵)
8 sticksstones19.7 . . 3 𝐺 = (𝑏𝐵 ↦ (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍𝑦))))
91, 2, 3, 4, 5, 8sticksstones17 40571 . 2 (𝜑𝐺:𝐵𝐴)
108a1i 11 . . . . 5 ((𝜑𝑐𝐴) → 𝐺 = (𝑏𝐵 ↦ (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍𝑦)))))
11 simplr 767 . . . . . . 7 ((((𝜑𝑐𝐴) ∧ 𝑏 = (𝐹𝑐)) ∧ 𝑦 ∈ (1...𝐾)) → 𝑏 = (𝐹𝑐))
1211fveq1d 6844 . . . . . 6 ((((𝜑𝑐𝐴) ∧ 𝑏 = (𝐹𝑐)) ∧ 𝑦 ∈ (1...𝐾)) → (𝑏‘(𝑍𝑦)) = ((𝐹𝑐)‘(𝑍𝑦)))
1312mpteq2dva 5205 . . . . 5 (((𝜑𝑐𝐴) ∧ 𝑏 = (𝐹𝑐)) → (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍𝑦))) = (𝑦 ∈ (1...𝐾) ↦ ((𝐹𝑐)‘(𝑍𝑦))))
147ffvelcdmda 7035 . . . . 5 ((𝜑𝑐𝐴) → (𝐹𝑐) ∈ 𝐵)
15 fzfid 13878 . . . . . 6 ((𝜑𝑐𝐴) → (1...𝐾) ∈ Fin)
1615mptexd 7174 . . . . 5 ((𝜑𝑐𝐴) → (𝑦 ∈ (1...𝐾) ↦ ((𝐹𝑐)‘(𝑍𝑦))) ∈ V)
1710, 13, 14, 16fvmptd 6955 . . . 4 ((𝜑𝑐𝐴) → (𝐺‘(𝐹𝑐)) = (𝑦 ∈ (1...𝐾) ↦ ((𝐹𝑐)‘(𝑍𝑦))))
186a1i 11 . . . . . . . . 9 ((𝜑𝑐𝐴𝑦 ∈ (1...𝐾)) → 𝐹 = (𝑎𝐴 ↦ (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥)))))
1918fveq1d 6844 . . . . . . . 8 ((𝜑𝑐𝐴𝑦 ∈ (1...𝐾)) → (𝐹𝑐) = ((𝑎𝐴 ↦ (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))))‘𝑐))
2019fveq1d 6844 . . . . . . 7 ((𝜑𝑐𝐴𝑦 ∈ (1...𝐾)) → ((𝐹𝑐)‘(𝑍𝑦)) = (((𝑎𝐴 ↦ (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))))‘𝑐)‘(𝑍𝑦)))
21203expa 1118 . . . . . 6 (((𝜑𝑐𝐴) ∧ 𝑦 ∈ (1...𝐾)) → ((𝐹𝑐)‘(𝑍𝑦)) = (((𝑎𝐴 ↦ (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))))‘𝑐)‘(𝑍𝑦)))
2221mpteq2dva 5205 . . . . 5 ((𝜑𝑐𝐴) → (𝑦 ∈ (1...𝐾) ↦ ((𝐹𝑐)‘(𝑍𝑦))) = (𝑦 ∈ (1...𝐾) ↦ (((𝑎𝐴 ↦ (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))))‘𝑐)‘(𝑍𝑦))))
23 eqidd 2737 . . . . . . . . 9 (((𝜑𝑐𝐴) ∧ 𝑦 ∈ (1...𝐾)) → (𝑎𝐴 ↦ (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥)))) = (𝑎𝐴 ↦ (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥)))))
24 simplr 767 . . . . . . . . . . 11 (((((𝜑𝑐𝐴) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑎 = 𝑐) ∧ 𝑥𝑆) → 𝑎 = 𝑐)
2524fveq1d 6844 . . . . . . . . . 10 (((((𝜑𝑐𝐴) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑎 = 𝑐) ∧ 𝑥𝑆) → (𝑎‘(𝑍𝑥)) = (𝑐‘(𝑍𝑥)))
2625mpteq2dva 5205 . . . . . . . . 9 ((((𝜑𝑐𝐴) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑎 = 𝑐) → (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))) = (𝑥𝑆 ↦ (𝑐‘(𝑍𝑥))))
27 simplr 767 . . . . . . . . 9 (((𝜑𝑐𝐴) ∧ 𝑦 ∈ (1...𝐾)) → 𝑐𝐴)
28 fzfid 13878 . . . . . . . . . . . . 13 (𝜑 → (1...𝐾) ∈ Fin)
29 f1oenfi 9126 . . . . . . . . . . . . . . 15 (((1...𝐾) ∈ Fin ∧ 𝑍:(1...𝐾)–1-1-onto𝑆) → (1...𝐾) ≈ 𝑆)
3028, 5, 29syl2anc 584 . . . . . . . . . . . . . 14 (𝜑 → (1...𝐾) ≈ 𝑆)
3130ensymd 8945 . . . . . . . . . . . . 13 (𝜑𝑆 ≈ (1...𝐾))
32 enfii 9133 . . . . . . . . . . . . 13 (((1...𝐾) ∈ Fin ∧ 𝑆 ≈ (1...𝐾)) → 𝑆 ∈ Fin)
3328, 31, 32syl2anc 584 . . . . . . . . . . . 12 (𝜑𝑆 ∈ Fin)
3433adantr 481 . . . . . . . . . . 11 ((𝜑𝑐𝐴) → 𝑆 ∈ Fin)
3534adantr 481 . . . . . . . . . 10 (((𝜑𝑐𝐴) ∧ 𝑦 ∈ (1...𝐾)) → 𝑆 ∈ Fin)
3635mptexd 7174 . . . . . . . . 9 (((𝜑𝑐𝐴) ∧ 𝑦 ∈ (1...𝐾)) → (𝑥𝑆 ↦ (𝑐‘(𝑍𝑥))) ∈ V)
3723, 26, 27, 36fvmptd 6955 . . . . . . . 8 (((𝜑𝑐𝐴) ∧ 𝑦 ∈ (1...𝐾)) → ((𝑎𝐴 ↦ (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))))‘𝑐) = (𝑥𝑆 ↦ (𝑐‘(𝑍𝑥))))
3837fveq1d 6844 . . . . . . 7 (((𝜑𝑐𝐴) ∧ 𝑦 ∈ (1...𝐾)) → (((𝑎𝐴 ↦ (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))))‘𝑐)‘(𝑍𝑦)) = ((𝑥𝑆 ↦ (𝑐‘(𝑍𝑥)))‘(𝑍𝑦)))
3938mpteq2dva 5205 . . . . . 6 ((𝜑𝑐𝐴) → (𝑦 ∈ (1...𝐾) ↦ (((𝑎𝐴 ↦ (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))))‘𝑐)‘(𝑍𝑦))) = (𝑦 ∈ (1...𝐾) ↦ ((𝑥𝑆 ↦ (𝑐‘(𝑍𝑥)))‘(𝑍𝑦))))
40 eqidd 2737 . . . . . . . . 9 (((𝜑𝑐𝐴) ∧ 𝑦 ∈ (1...𝐾)) → (𝑥𝑆 ↦ (𝑐‘(𝑍𝑥))) = (𝑥𝑆 ↦ (𝑐‘(𝑍𝑥))))
41 simpr 485 . . . . . . . . . . 11 ((((𝜑𝑐𝐴) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑥 = (𝑍𝑦)) → 𝑥 = (𝑍𝑦))
4241fveq2d 6846 . . . . . . . . . 10 ((((𝜑𝑐𝐴) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑥 = (𝑍𝑦)) → (𝑍𝑥) = (𝑍‘(𝑍𝑦)))
4342fveq2d 6846 . . . . . . . . 9 ((((𝜑𝑐𝐴) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑥 = (𝑍𝑦)) → (𝑐‘(𝑍𝑥)) = (𝑐‘(𝑍‘(𝑍𝑦))))
44 f1of 6784 . . . . . . . . . . . 12 (𝑍:(1...𝐾)–1-1-onto𝑆𝑍:(1...𝐾)⟶𝑆)
455, 44syl 17 . . . . . . . . . . 11 (𝜑𝑍:(1...𝐾)⟶𝑆)
4645adantr 481 . . . . . . . . . 10 ((𝜑𝑐𝐴) → 𝑍:(1...𝐾)⟶𝑆)
4746ffvelcdmda 7035 . . . . . . . . 9 (((𝜑𝑐𝐴) ∧ 𝑦 ∈ (1...𝐾)) → (𝑍𝑦) ∈ 𝑆)
48 fvexd 6857 . . . . . . . . 9 (((𝜑𝑐𝐴) ∧ 𝑦 ∈ (1...𝐾)) → (𝑐‘(𝑍‘(𝑍𝑦))) ∈ V)
4940, 43, 47, 48fvmptd 6955 . . . . . . . 8 (((𝜑𝑐𝐴) ∧ 𝑦 ∈ (1...𝐾)) → ((𝑥𝑆 ↦ (𝑐‘(𝑍𝑥)))‘(𝑍𝑦)) = (𝑐‘(𝑍‘(𝑍𝑦))))
5049mpteq2dva 5205 . . . . . . 7 ((𝜑𝑐𝐴) → (𝑦 ∈ (1...𝐾) ↦ ((𝑥𝑆 ↦ (𝑐‘(𝑍𝑥)))‘(𝑍𝑦))) = (𝑦 ∈ (1...𝐾) ↦ (𝑐‘(𝑍‘(𝑍𝑦)))))
515ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝑐𝐴) ∧ 𝑦 ∈ (1...𝐾)) → 𝑍:(1...𝐾)–1-1-onto𝑆)
52 simpr 485 . . . . . . . . . . 11 (((𝜑𝑐𝐴) ∧ 𝑦 ∈ (1...𝐾)) → 𝑦 ∈ (1...𝐾))
53 f1ocnvfv1 7222 . . . . . . . . . . 11 ((𝑍:(1...𝐾)–1-1-onto𝑆𝑦 ∈ (1...𝐾)) → (𝑍‘(𝑍𝑦)) = 𝑦)
5451, 52, 53syl2anc 584 . . . . . . . . . 10 (((𝜑𝑐𝐴) ∧ 𝑦 ∈ (1...𝐾)) → (𝑍‘(𝑍𝑦)) = 𝑦)
5554fveq2d 6846 . . . . . . . . 9 (((𝜑𝑐𝐴) ∧ 𝑦 ∈ (1...𝐾)) → (𝑐‘(𝑍‘(𝑍𝑦))) = (𝑐𝑦))
5655mpteq2dva 5205 . . . . . . . 8 ((𝜑𝑐𝐴) → (𝑦 ∈ (1...𝐾) ↦ (𝑐‘(𝑍‘(𝑍𝑦)))) = (𝑦 ∈ (1...𝐾) ↦ (𝑐𝑦)))
57 simpr 485 . . . . . . . . . . . . . 14 ((𝜑𝑐𝐴) → 𝑐𝐴)
583a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑐𝐴) → 𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔𝑖) = 𝑁)})
5957, 58eleqtrd 2840 . . . . . . . . . . . . 13 ((𝜑𝑐𝐴) → 𝑐 ∈ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔𝑖) = 𝑁)})
60 vex 3449 . . . . . . . . . . . . . 14 𝑐 ∈ V
61 feq1 6649 . . . . . . . . . . . . . . 15 (𝑔 = 𝑐 → (𝑔:(1...𝐾)⟶ℕ0𝑐:(1...𝐾)⟶ℕ0))
62 simpl 483 . . . . . . . . . . . . . . . . . 18 ((𝑔 = 𝑐𝑖 ∈ (1...𝐾)) → 𝑔 = 𝑐)
6362fveq1d 6844 . . . . . . . . . . . . . . . . 17 ((𝑔 = 𝑐𝑖 ∈ (1...𝐾)) → (𝑔𝑖) = (𝑐𝑖))
6463sumeq2dv 15588 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑐 → Σ𝑖 ∈ (1...𝐾)(𝑔𝑖) = Σ𝑖 ∈ (1...𝐾)(𝑐𝑖))
6564eqeq1d 2738 . . . . . . . . . . . . . . 15 (𝑔 = 𝑐 → (Σ𝑖 ∈ (1...𝐾)(𝑔𝑖) = 𝑁 ↔ Σ𝑖 ∈ (1...𝐾)(𝑐𝑖) = 𝑁))
6661, 65anbi12d 631 . . . . . . . . . . . . . 14 (𝑔 = 𝑐 → ((𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔𝑖) = 𝑁) ↔ (𝑐:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑐𝑖) = 𝑁)))
6760, 66elab 3630 . . . . . . . . . . . . 13 (𝑐 ∈ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔𝑖) = 𝑁)} ↔ (𝑐:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑐𝑖) = 𝑁))
6859, 67sylib 217 . . . . . . . . . . . 12 ((𝜑𝑐𝐴) → (𝑐:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑐𝑖) = 𝑁))
6968simpld 495 . . . . . . . . . . 11 ((𝜑𝑐𝐴) → 𝑐:(1...𝐾)⟶ℕ0)
70 ffn 6668 . . . . . . . . . . 11 (𝑐:(1...𝐾)⟶ℕ0𝑐 Fn (1...𝐾))
7169, 70syl 17 . . . . . . . . . 10 ((𝜑𝑐𝐴) → 𝑐 Fn (1...𝐾))
72 dffn5 6901 . . . . . . . . . 10 (𝑐 Fn (1...𝐾) ↔ 𝑐 = (𝑦 ∈ (1...𝐾) ↦ (𝑐𝑦)))
7371, 72sylib 217 . . . . . . . . 9 ((𝜑𝑐𝐴) → 𝑐 = (𝑦 ∈ (1...𝐾) ↦ (𝑐𝑦)))
7473eqcomd 2742 . . . . . . . 8 ((𝜑𝑐𝐴) → (𝑦 ∈ (1...𝐾) ↦ (𝑐𝑦)) = 𝑐)
7556, 74eqtrd 2776 . . . . . . 7 ((𝜑𝑐𝐴) → (𝑦 ∈ (1...𝐾) ↦ (𝑐‘(𝑍‘(𝑍𝑦)))) = 𝑐)
7650, 75eqtrd 2776 . . . . . 6 ((𝜑𝑐𝐴) → (𝑦 ∈ (1...𝐾) ↦ ((𝑥𝑆 ↦ (𝑐‘(𝑍𝑥)))‘(𝑍𝑦))) = 𝑐)
7739, 76eqtrd 2776 . . . . 5 ((𝜑𝑐𝐴) → (𝑦 ∈ (1...𝐾) ↦ (((𝑎𝐴 ↦ (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))))‘𝑐)‘(𝑍𝑦))) = 𝑐)
7822, 77eqtrd 2776 . . . 4 ((𝜑𝑐𝐴) → (𝑦 ∈ (1...𝐾) ↦ ((𝐹𝑐)‘(𝑍𝑦))) = 𝑐)
7917, 78eqtrd 2776 . . 3 ((𝜑𝑐𝐴) → (𝐺‘(𝐹𝑐)) = 𝑐)
8079ralrimiva 3143 . 2 (𝜑 → ∀𝑐𝐴 (𝐺‘(𝐹𝑐)) = 𝑐)
816a1i 11 . . . . 5 ((𝜑𝑑𝐵) → 𝐹 = (𝑎𝐴 ↦ (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥)))))
82 simplr 767 . . . . . . 7 ((((𝜑𝑑𝐵) ∧ 𝑎 = (𝐺𝑑)) ∧ 𝑥𝑆) → 𝑎 = (𝐺𝑑))
8382fveq1d 6844 . . . . . 6 ((((𝜑𝑑𝐵) ∧ 𝑎 = (𝐺𝑑)) ∧ 𝑥𝑆) → (𝑎‘(𝑍𝑥)) = ((𝐺𝑑)‘(𝑍𝑥)))
8483mpteq2dva 5205 . . . . 5 (((𝜑𝑑𝐵) ∧ 𝑎 = (𝐺𝑑)) → (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))) = (𝑥𝑆 ↦ ((𝐺𝑑)‘(𝑍𝑥))))
859ffvelcdmda 7035 . . . . 5 ((𝜑𝑑𝐵) → (𝐺𝑑) ∈ 𝐴)
8633adantr 481 . . . . . 6 ((𝜑𝑑𝐵) → 𝑆 ∈ Fin)
8786mptexd 7174 . . . . 5 ((𝜑𝑑𝐵) → (𝑥𝑆 ↦ ((𝐺𝑑)‘(𝑍𝑥))) ∈ V)
8881, 84, 85, 87fvmptd 6955 . . . 4 ((𝜑𝑑𝐵) → (𝐹‘(𝐺𝑑)) = (𝑥𝑆 ↦ ((𝐺𝑑)‘(𝑍𝑥))))
898a1i 11 . . . . . . . 8 (((𝜑𝑑𝐵) ∧ 𝑥𝑆) → 𝐺 = (𝑏𝐵 ↦ (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍𝑦)))))
90 simplr 767 . . . . . . . . . 10 (((((𝜑𝑑𝐵) ∧ 𝑥𝑆) ∧ 𝑏 = 𝑑) ∧ 𝑦 ∈ (1...𝐾)) → 𝑏 = 𝑑)
9190fveq1d 6844 . . . . . . . . 9 (((((𝜑𝑑𝐵) ∧ 𝑥𝑆) ∧ 𝑏 = 𝑑) ∧ 𝑦 ∈ (1...𝐾)) → (𝑏‘(𝑍𝑦)) = (𝑑‘(𝑍𝑦)))
9291mpteq2dva 5205 . . . . . . . 8 ((((𝜑𝑑𝐵) ∧ 𝑥𝑆) ∧ 𝑏 = 𝑑) → (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍𝑦))) = (𝑦 ∈ (1...𝐾) ↦ (𝑑‘(𝑍𝑦))))
93 simplr 767 . . . . . . . 8 (((𝜑𝑑𝐵) ∧ 𝑥𝑆) → 𝑑𝐵)
94 fzfid 13878 . . . . . . . . 9 (((𝜑𝑑𝐵) ∧ 𝑥𝑆) → (1...𝐾) ∈ Fin)
9594mptexd 7174 . . . . . . . 8 (((𝜑𝑑𝐵) ∧ 𝑥𝑆) → (𝑦 ∈ (1...𝐾) ↦ (𝑑‘(𝑍𝑦))) ∈ V)
9689, 92, 93, 95fvmptd 6955 . . . . . . 7 (((𝜑𝑑𝐵) ∧ 𝑥𝑆) → (𝐺𝑑) = (𝑦 ∈ (1...𝐾) ↦ (𝑑‘(𝑍𝑦))))
9796fveq1d 6844 . . . . . 6 (((𝜑𝑑𝐵) ∧ 𝑥𝑆) → ((𝐺𝑑)‘(𝑍𝑥)) = ((𝑦 ∈ (1...𝐾) ↦ (𝑑‘(𝑍𝑦)))‘(𝑍𝑥)))
9897mpteq2dva 5205 . . . . 5 ((𝜑𝑑𝐵) → (𝑥𝑆 ↦ ((𝐺𝑑)‘(𝑍𝑥))) = (𝑥𝑆 ↦ ((𝑦 ∈ (1...𝐾) ↦ (𝑑‘(𝑍𝑦)))‘(𝑍𝑥))))
99 eqidd 2737 . . . . . . . 8 (((𝜑𝑑𝐵) ∧ 𝑥𝑆) → (𝑦 ∈ (1...𝐾) ↦ (𝑑‘(𝑍𝑦))) = (𝑦 ∈ (1...𝐾) ↦ (𝑑‘(𝑍𝑦))))
100 simpr 485 . . . . . . . . . 10 ((((𝜑𝑑𝐵) ∧ 𝑥𝑆) ∧ 𝑦 = (𝑍𝑥)) → 𝑦 = (𝑍𝑥))
101100fveq2d 6846 . . . . . . . . 9 ((((𝜑𝑑𝐵) ∧ 𝑥𝑆) ∧ 𝑦 = (𝑍𝑥)) → (𝑍𝑦) = (𝑍‘(𝑍𝑥)))
102101fveq2d 6846 . . . . . . . 8 ((((𝜑𝑑𝐵) ∧ 𝑥𝑆) ∧ 𝑦 = (𝑍𝑥)) → (𝑑‘(𝑍𝑦)) = (𝑑‘(𝑍‘(𝑍𝑥))))
103 f1ocnv 6796 . . . . . . . . . . . 12 (𝑍:(1...𝐾)–1-1-onto𝑆𝑍:𝑆1-1-onto→(1...𝐾))
1045, 103syl 17 . . . . . . . . . . 11 (𝜑𝑍:𝑆1-1-onto→(1...𝐾))
105 f1of 6784 . . . . . . . . . . 11 (𝑍:𝑆1-1-onto→(1...𝐾) → 𝑍:𝑆⟶(1...𝐾))
106104, 105syl 17 . . . . . . . . . 10 (𝜑𝑍:𝑆⟶(1...𝐾))
107106adantr 481 . . . . . . . . 9 ((𝜑𝑑𝐵) → 𝑍:𝑆⟶(1...𝐾))
108107ffvelcdmda 7035 . . . . . . . 8 (((𝜑𝑑𝐵) ∧ 𝑥𝑆) → (𝑍𝑥) ∈ (1...𝐾))
109 fvexd 6857 . . . . . . . 8 (((𝜑𝑑𝐵) ∧ 𝑥𝑆) → (𝑑‘(𝑍‘(𝑍𝑥))) ∈ V)
11099, 102, 108, 109fvmptd 6955 . . . . . . 7 (((𝜑𝑑𝐵) ∧ 𝑥𝑆) → ((𝑦 ∈ (1...𝐾) ↦ (𝑑‘(𝑍𝑦)))‘(𝑍𝑥)) = (𝑑‘(𝑍‘(𝑍𝑥))))
111110mpteq2dva 5205 . . . . . 6 ((𝜑𝑑𝐵) → (𝑥𝑆 ↦ ((𝑦 ∈ (1...𝐾) ↦ (𝑑‘(𝑍𝑦)))‘(𝑍𝑥))) = (𝑥𝑆 ↦ (𝑑‘(𝑍‘(𝑍𝑥)))))
1125ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑑𝐵) ∧ 𝑥𝑆) → 𝑍:(1...𝐾)–1-1-onto𝑆)
113 simpr 485 . . . . . . . . . 10 (((𝜑𝑑𝐵) ∧ 𝑥𝑆) → 𝑥𝑆)
114 f1ocnvfv2 7223 . . . . . . . . . 10 ((𝑍:(1...𝐾)–1-1-onto𝑆𝑥𝑆) → (𝑍‘(𝑍𝑥)) = 𝑥)
115112, 113, 114syl2anc 584 . . . . . . . . 9 (((𝜑𝑑𝐵) ∧ 𝑥𝑆) → (𝑍‘(𝑍𝑥)) = 𝑥)
116115fveq2d 6846 . . . . . . . 8 (((𝜑𝑑𝐵) ∧ 𝑥𝑆) → (𝑑‘(𝑍‘(𝑍𝑥))) = (𝑑𝑥))
117116mpteq2dva 5205 . . . . . . 7 ((𝜑𝑑𝐵) → (𝑥𝑆 ↦ (𝑑‘(𝑍‘(𝑍𝑥)))) = (𝑥𝑆 ↦ (𝑑𝑥)))
118 simpr 485 . . . . . . . . . . . . 13 ((𝜑𝑑𝐵) → 𝑑𝐵)
1194a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑑𝐵) → 𝐵 = { ∣ (:𝑆⟶ℕ0 ∧ Σ𝑖𝑆 (𝑖) = 𝑁)})
120118, 119eleqtrd 2840 . . . . . . . . . . . 12 ((𝜑𝑑𝐵) → 𝑑 ∈ { ∣ (:𝑆⟶ℕ0 ∧ Σ𝑖𝑆 (𝑖) = 𝑁)})
121 vex 3449 . . . . . . . . . . . . 13 𝑑 ∈ V
122 feq1 6649 . . . . . . . . . . . . . 14 ( = 𝑑 → (:𝑆⟶ℕ0𝑑:𝑆⟶ℕ0))
123 simpl 483 . . . . . . . . . . . . . . . . 17 (( = 𝑑𝑖𝑆) → = 𝑑)
124123fveq1d 6844 . . . . . . . . . . . . . . . 16 (( = 𝑑𝑖𝑆) → (𝑖) = (𝑑𝑖))
125124sumeq2dv 15588 . . . . . . . . . . . . . . 15 ( = 𝑑 → Σ𝑖𝑆 (𝑖) = Σ𝑖𝑆 (𝑑𝑖))
126125eqeq1d 2738 . . . . . . . . . . . . . 14 ( = 𝑑 → (Σ𝑖𝑆 (𝑖) = 𝑁 ↔ Σ𝑖𝑆 (𝑑𝑖) = 𝑁))
127122, 126anbi12d 631 . . . . . . . . . . . . 13 ( = 𝑑 → ((:𝑆⟶ℕ0 ∧ Σ𝑖𝑆 (𝑖) = 𝑁) ↔ (𝑑:𝑆⟶ℕ0 ∧ Σ𝑖𝑆 (𝑑𝑖) = 𝑁)))
128121, 127elab 3630 . . . . . . . . . . . 12 (𝑑 ∈ { ∣ (:𝑆⟶ℕ0 ∧ Σ𝑖𝑆 (𝑖) = 𝑁)} ↔ (𝑑:𝑆⟶ℕ0 ∧ Σ𝑖𝑆 (𝑑𝑖) = 𝑁))
129120, 128sylib 217 . . . . . . . . . . 11 ((𝜑𝑑𝐵) → (𝑑:𝑆⟶ℕ0 ∧ Σ𝑖𝑆 (𝑑𝑖) = 𝑁))
130129simpld 495 . . . . . . . . . 10 ((𝜑𝑑𝐵) → 𝑑:𝑆⟶ℕ0)
131 ffn 6668 . . . . . . . . . 10 (𝑑:𝑆⟶ℕ0𝑑 Fn 𝑆)
132130, 131syl 17 . . . . . . . . 9 ((𝜑𝑑𝐵) → 𝑑 Fn 𝑆)
133 dffn5 6901 . . . . . . . . 9 (𝑑 Fn 𝑆𝑑 = (𝑥𝑆 ↦ (𝑑𝑥)))
134132, 133sylib 217 . . . . . . . 8 ((𝜑𝑑𝐵) → 𝑑 = (𝑥𝑆 ↦ (𝑑𝑥)))
135134eqcomd 2742 . . . . . . 7 ((𝜑𝑑𝐵) → (𝑥𝑆 ↦ (𝑑𝑥)) = 𝑑)
136117, 135eqtrd 2776 . . . . . 6 ((𝜑𝑑𝐵) → (𝑥𝑆 ↦ (𝑑‘(𝑍‘(𝑍𝑥)))) = 𝑑)
137111, 136eqtrd 2776 . . . . 5 ((𝜑𝑑𝐵) → (𝑥𝑆 ↦ ((𝑦 ∈ (1...𝐾) ↦ (𝑑‘(𝑍𝑦)))‘(𝑍𝑥))) = 𝑑)
13898, 137eqtrd 2776 . . . 4 ((𝜑𝑑𝐵) → (𝑥𝑆 ↦ ((𝐺𝑑)‘(𝑍𝑥))) = 𝑑)
13988, 138eqtrd 2776 . . 3 ((𝜑𝑑𝐵) → (𝐹‘(𝐺𝑑)) = 𝑑)
140139ralrimiva 3143 . 2 (𝜑 → ∀𝑑𝐵 (𝐹‘(𝐺𝑑)) = 𝑑)
1417, 9, 80, 1402fvidf1od 7244 1 (𝜑𝐹:𝐴1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  {cab 2713  Vcvv 3445   class class class wbr 5105  cmpt 5188  ccnv 5632   Fn wfn 6491  wf 6492  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  cen 8880  Fincfn 8883  1c1 11052  0cn0 12413  ...cfz 13424  Σcsu 15570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571
This theorem is referenced by:  sticksstones20  40574
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