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Theorem f1prex 6493
Description: Relate a one-to-one function with a pair as domain and two different variables. (Contributed by Thierry Arnoux, 12-Jul-2020.)
Hypotheses
Ref Expression
f1prex.1 (𝑥 = (𝑓𝐴) → (𝜓𝜒))
f1prex.2 (𝑦 = (𝑓𝐵) → (𝜒𝜑))
Assertion
Ref Expression
f1prex ((𝐴𝑉𝐵𝑊𝐴𝐵) → (∃𝑓(𝑓:{𝐴, 𝐵}–1-1𝐷𝜑) ↔ ∃𝑥𝐷𝑦𝐷 (𝑥𝑦𝜓)))
Distinct variable groups:   𝐴,𝑓,𝑥,𝑦   𝐵,𝑓,𝑥,𝑦   𝐷,𝑓,𝑥,𝑦   𝑓,𝑉,𝑥,𝑦   𝑓,𝑊,𝑥,𝑦   𝜒,𝑥   𝜓,𝑓   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑓)   𝜓(𝑥,𝑦)   𝜒(𝑦,𝑓)

Proof of Theorem f1prex
StepHypRef Expression
1 simpl1 1062 . . . . . . 7 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)) → 𝐴𝑉)
2 simpl2 1063 . . . . . . 7 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)) → 𝐵𝑊)
3 simprl 793 . . . . . . . 8 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)) → 𝑓:{𝐴, 𝐵}–1-1𝐷)
4 f1f 6058 . . . . . . . 8 (𝑓:{𝐴, 𝐵}–1-1𝐷𝑓:{𝐴, 𝐵}⟶𝐷)
53, 4syl 17 . . . . . . 7 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)) → 𝑓:{𝐴, 𝐵}⟶𝐷)
6 fpr2g 6429 . . . . . . . . 9 ((𝐴𝑉𝐵𝑊) → (𝑓:{𝐴, 𝐵}⟶𝐷 ↔ ((𝑓𝐴) ∈ 𝐷 ∧ (𝑓𝐵) ∈ 𝐷𝑓 = {⟨𝐴, (𝑓𝐴)⟩, ⟨𝐵, (𝑓𝐵)⟩})))
76biimpa 501 . . . . . . . 8 (((𝐴𝑉𝐵𝑊) ∧ 𝑓:{𝐴, 𝐵}⟶𝐷) → ((𝑓𝐴) ∈ 𝐷 ∧ (𝑓𝐵) ∈ 𝐷𝑓 = {⟨𝐴, (𝑓𝐴)⟩, ⟨𝐵, (𝑓𝐵)⟩}))
87simp1d 1071 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ 𝑓:{𝐴, 𝐵}⟶𝐷) → (𝑓𝐴) ∈ 𝐷)
91, 2, 5, 8syl21anc 1322 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)) → (𝑓𝐴) ∈ 𝐷)
107simp2d 1072 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ 𝑓:{𝐴, 𝐵}⟶𝐷) → (𝑓𝐵) ∈ 𝐷)
111, 2, 5, 10syl21anc 1322 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)) → (𝑓𝐵) ∈ 𝐷)
12 prid1g 4265 . . . . . . . . . 10 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
131, 12syl 17 . . . . . . . . 9 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)) → 𝐴 ∈ {𝐴, 𝐵})
14 prid2g 4266 . . . . . . . . . 10 (𝐵𝑊𝐵 ∈ {𝐴, 𝐵})
152, 14syl 17 . . . . . . . . 9 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)) → 𝐵 ∈ {𝐴, 𝐵})
1613, 15jca 554 . . . . . . . 8 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)) → (𝐴 ∈ {𝐴, 𝐵} ∧ 𝐵 ∈ {𝐴, 𝐵}))
17 simpl3 1064 . . . . . . . 8 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)) → 𝐴𝐵)
18 f1veqaeq 6468 . . . . . . . . . 10 ((𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝐴 ∈ {𝐴, 𝐵} ∧ 𝐵 ∈ {𝐴, 𝐵})) → ((𝑓𝐴) = (𝑓𝐵) → 𝐴 = 𝐵))
1918necon3d 2811 . . . . . . . . 9 ((𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝐴 ∈ {𝐴, 𝐵} ∧ 𝐵 ∈ {𝐴, 𝐵})) → (𝐴𝐵 → (𝑓𝐴) ≠ (𝑓𝐵)))
2019imp 445 . . . . . . . 8 (((𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝐴 ∈ {𝐴, 𝐵} ∧ 𝐵 ∈ {𝐴, 𝐵})) ∧ 𝐴𝐵) → (𝑓𝐴) ≠ (𝑓𝐵))
213, 16, 17, 20syl21anc 1322 . . . . . . 7 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)) → (𝑓𝐴) ≠ (𝑓𝐵))
22 simprr 795 . . . . . . 7 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)) → 𝜑)
2321, 22jca 554 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)) → ((𝑓𝐴) ≠ (𝑓𝐵) ∧ 𝜑))
24 neeq1 2852 . . . . . . . 8 (𝑥 = (𝑓𝐴) → (𝑥𝑦 ↔ (𝑓𝐴) ≠ 𝑦))
25 f1prex.1 . . . . . . . 8 (𝑥 = (𝑓𝐴) → (𝜓𝜒))
2624, 25anbi12d 746 . . . . . . 7 (𝑥 = (𝑓𝐴) → ((𝑥𝑦𝜓) ↔ ((𝑓𝐴) ≠ 𝑦𝜒)))
27 neeq2 2853 . . . . . . . 8 (𝑦 = (𝑓𝐵) → ((𝑓𝐴) ≠ 𝑦 ↔ (𝑓𝐴) ≠ (𝑓𝐵)))
28 f1prex.2 . . . . . . . 8 (𝑦 = (𝑓𝐵) → (𝜒𝜑))
2927, 28anbi12d 746 . . . . . . 7 (𝑦 = (𝑓𝐵) → (((𝑓𝐴) ≠ 𝑦𝜒) ↔ ((𝑓𝐴) ≠ (𝑓𝐵) ∧ 𝜑)))
3026, 29rspc2ev 3308 . . . . . 6 (((𝑓𝐴) ∈ 𝐷 ∧ (𝑓𝐵) ∈ 𝐷 ∧ ((𝑓𝐴) ≠ (𝑓𝐵) ∧ 𝜑)) → ∃𝑥𝐷𝑦𝐷 (𝑥𝑦𝜓))
319, 11, 23, 30syl3anc 1323 . . . . 5 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)) → ∃𝑥𝐷𝑦𝐷 (𝑥𝑦𝜓))
3231ex 450 . . . 4 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ((𝑓:{𝐴, 𝐵}–1-1𝐷𝜑) → ∃𝑥𝐷𝑦𝐷 (𝑥𝑦𝜓)))
3332exlimdv 1858 . . 3 ((𝐴𝑉𝐵𝑊𝐴𝐵) → (∃𝑓(𝑓:{𝐴, 𝐵}–1-1𝐷𝜑) → ∃𝑥𝐷𝑦𝐷 (𝑥𝑦𝜓)))
3433imp 445 . 2 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ ∃𝑓(𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)) → ∃𝑥𝐷𝑦𝐷 (𝑥𝑦𝜓))
35 simpll1 1098 . . . . . . . . . . 11 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → 𝐴𝑉)
36 simplrl 799 . . . . . . . . . . 11 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → 𝑥𝐷)
3735, 36jca 554 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → (𝐴𝑉𝑥𝐷))
38 simpll2 1099 . . . . . . . . . . 11 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → 𝐵𝑊)
39 simplrr 800 . . . . . . . . . . 11 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → 𝑦𝐷)
4038, 39jca 554 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → (𝐵𝑊𝑦𝐷))
41 simpll3 1100 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → 𝐴𝐵)
42 simprl 793 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → 𝑥𝑦)
43 f1oprg 6138 . . . . . . . . . . 11 (((𝐴𝑉𝑥𝐷) ∧ (𝐵𝑊𝑦𝐷)) → ((𝐴𝐵𝑥𝑦) → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}–1-1-onto→{𝑥, 𝑦}))
4443imp 445 . . . . . . . . . 10 ((((𝐴𝑉𝑥𝐷) ∧ (𝐵𝑊𝑦𝐷)) ∧ (𝐴𝐵𝑥𝑦)) → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}–1-1-onto→{𝑥, 𝑦})
4537, 40, 41, 42, 44syl22anc 1324 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}–1-1-onto→{𝑥, 𝑦})
46 f1of1 6093 . . . . . . . . 9 ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}–1-1-onto→{𝑥, 𝑦} → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}–1-1→{𝑥, 𝑦})
4745, 46syl 17 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}–1-1→{𝑥, 𝑦})
48 prssi 4321 . . . . . . . . 9 ((𝑥𝐷𝑦𝐷) → {𝑥, 𝑦} ⊆ 𝐷)
4936, 39, 48syl2anc 692 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → {𝑥, 𝑦} ⊆ 𝐷)
50 f1ss 6063 . . . . . . . 8 (({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}–1-1→{𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝐷) → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}–1-1𝐷)
5147, 49, 50syl2anc 692 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}–1-1𝐷)
52 fvpr1g 6412 . . . . . . . . 9 ((𝐴𝑉𝑥𝐷𝐴𝐵) → ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐴) = 𝑥)
5352eqcomd 2627 . . . . . . . 8 ((𝐴𝑉𝑥𝐷𝐴𝐵) → 𝑥 = ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐴))
5435, 36, 41, 53syl3anc 1323 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → 𝑥 = ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐴))
55 fvpr2g 6413 . . . . . . . . 9 ((𝐵𝑊𝑦𝐷𝐴𝐵) → ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐵) = 𝑦)
5655eqcomd 2627 . . . . . . . 8 ((𝐵𝑊𝑦𝐷𝐴𝐵) → 𝑦 = ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐵))
5738, 39, 41, 56syl3anc 1323 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → 𝑦 = ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐵))
58 prex 4870 . . . . . . . 8 {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} ∈ V
59 f1eq1 6053 . . . . . . . . 9 (𝑓 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} → (𝑓:{𝐴, 𝐵}–1-1𝐷 ↔ {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}–1-1𝐷))
60 fveq1 6147 . . . . . . . . . . 11 (𝑓 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} → (𝑓𝐴) = ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐴))
6160eqeq2d 2631 . . . . . . . . . 10 (𝑓 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} → (𝑥 = (𝑓𝐴) ↔ 𝑥 = ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐴)))
62 fveq1 6147 . . . . . . . . . . 11 (𝑓 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} → (𝑓𝐵) = ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐵))
6362eqeq2d 2631 . . . . . . . . . 10 (𝑓 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} → (𝑦 = (𝑓𝐵) ↔ 𝑦 = ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐵)))
6461, 63anbi12d 746 . . . . . . . . 9 (𝑓 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} → ((𝑥 = (𝑓𝐴) ∧ 𝑦 = (𝑓𝐵)) ↔ (𝑥 = ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐴) ∧ 𝑦 = ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐵))))
6559, 64anbi12d 746 . . . . . . . 8 (𝑓 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} → ((𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = (𝑓𝐴) ∧ 𝑦 = (𝑓𝐵))) ↔ ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐴) ∧ 𝑦 = ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐵)))))
6658, 65spcev 3286 . . . . . . 7 (({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐴) ∧ 𝑦 = ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐵))) → ∃𝑓(𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = (𝑓𝐴) ∧ 𝑦 = (𝑓𝐵))))
6751, 54, 57, 66syl12anc 1321 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → ∃𝑓(𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = (𝑓𝐴) ∧ 𝑦 = (𝑓𝐵))))
68 simprl 793 . . . . . . . . 9 (((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = (𝑓𝐴) ∧ 𝑦 = (𝑓𝐵)))) → 𝑓:{𝐴, 𝐵}–1-1𝐷)
69 simplrr 800 . . . . . . . . . . 11 (((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = (𝑓𝐴) ∧ 𝑦 = (𝑓𝐵)))) → 𝜓)
70 simprrl 803 . . . . . . . . . . . 12 (((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = (𝑓𝐴) ∧ 𝑦 = (𝑓𝐵)))) → 𝑥 = (𝑓𝐴))
7170, 25syl 17 . . . . . . . . . . 11 (((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = (𝑓𝐴) ∧ 𝑦 = (𝑓𝐵)))) → (𝜓𝜒))
7269, 71mpbid 222 . . . . . . . . . 10 (((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = (𝑓𝐴) ∧ 𝑦 = (𝑓𝐵)))) → 𝜒)
73 simprrr 804 . . . . . . . . . . 11 (((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = (𝑓𝐴) ∧ 𝑦 = (𝑓𝐵)))) → 𝑦 = (𝑓𝐵))
7473, 28syl 17 . . . . . . . . . 10 (((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = (𝑓𝐴) ∧ 𝑦 = (𝑓𝐵)))) → (𝜒𝜑))
7572, 74mpbid 222 . . . . . . . . 9 (((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = (𝑓𝐴) ∧ 𝑦 = (𝑓𝐵)))) → 𝜑)
7668, 75jca 554 . . . . . . . 8 (((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = (𝑓𝐴) ∧ 𝑦 = (𝑓𝐵)))) → (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑))
7776ex 450 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → ((𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = (𝑓𝐴) ∧ 𝑦 = (𝑓𝐵))) → (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)))
7877eximdv 1843 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → (∃𝑓(𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = (𝑓𝐴) ∧ 𝑦 = (𝑓𝐵))) → ∃𝑓(𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)))
7967, 78mpd 15 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → ∃𝑓(𝑓:{𝐴, 𝐵}–1-1𝐷𝜑))
8079ex 450 . . . 4 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) → ((𝑥𝑦𝜓) → ∃𝑓(𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)))
8180rexlimdvva 3031 . . 3 ((𝐴𝑉𝐵𝑊𝐴𝐵) → (∃𝑥𝐷𝑦𝐷 (𝑥𝑦𝜓) → ∃𝑓(𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)))
8281imp 445 . 2 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ ∃𝑥𝐷𝑦𝐷 (𝑥𝑦𝜓)) → ∃𝑓(𝑓:{𝐴, 𝐵}–1-1𝐷𝜑))
8334, 82impbida 876 1 ((𝐴𝑉𝐵𝑊𝐴𝐵) → (∃𝑓(𝑓:{𝐴, 𝐵}–1-1𝐷𝜑) ↔ ∃𝑥𝐷𝑦𝐷 (𝑥𝑦𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wne 2790  wrex 2908  wss 3555  {cpr 4150  cop 4154  wf 5843  1-1wf1 5844  1-1-ontowf1o 5846  cfv 5847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855
This theorem is referenced by:  istrkg3ld  25260
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