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Theorem fpwrelmap 32744
Description: Define a canonical mapping between functions from 𝐴 into subsets of 𝐵 and the relations with domain 𝐴 and range within 𝐵. Note that the same relation is used in axdc2lem 10488 and marypha2lem1 9475. (Contributed by Thierry Arnoux, 28-Aug-2017.)
Hypotheses
Ref Expression
fpwrelmap.1 𝐴 ∈ V
fpwrelmap.2 𝐵 ∈ V
fpwrelmap.3 𝑀 = (𝑓 ∈ (𝒫 𝐵m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
Assertion
Ref Expression
fpwrelmap 𝑀:(𝒫 𝐵m 𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵)
Distinct variable groups:   𝑥,𝑓,𝑦,𝐴   𝐵,𝑓,𝑥,𝑦
Allowed substitution hints:   𝑀(𝑥,𝑦,𝑓)

Proof of Theorem fpwrelmap
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 fpwrelmap.3 . . 3 𝑀 = (𝑓 ∈ (𝒫 𝐵m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
2 fpwrelmap.1 . . . . . 6 𝐴 ∈ V
32a1i 11 . . . . 5 (𝑓 ∈ (𝒫 𝐵m 𝐴) → 𝐴 ∈ V)
4 simpr 484 . . . . . . . . 9 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑦 ∈ (𝑓𝑥))
5 elmapi 8889 . . . . . . . . . . 11 (𝑓 ∈ (𝒫 𝐵m 𝐴) → 𝑓:𝐴⟶𝒫 𝐵)
65ffvelcdmda 7104 . . . . . . . . . 10 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑥𝐴) → (𝑓𝑥) ∈ 𝒫 𝐵)
76adantr 480 . . . . . . . . 9 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → (𝑓𝑥) ∈ 𝒫 𝐵)
8 elelpwi 4610 . . . . . . . . 9 ((𝑦 ∈ (𝑓𝑥) ∧ (𝑓𝑥) ∈ 𝒫 𝐵) → 𝑦𝐵)
94, 7, 8syl2anc 584 . . . . . . . 8 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑦𝐵)
109ex 412 . . . . . . 7 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑥𝐴) → (𝑦 ∈ (𝑓𝑥) → 𝑦𝐵))
1110alrimiv 1927 . . . . . 6 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑥𝐴) → ∀𝑦(𝑦 ∈ (𝑓𝑥) → 𝑦𝐵))
12 abss 4063 . . . . . . 7 ({𝑦𝑦 ∈ (𝑓𝑥)} ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ (𝑓𝑥) → 𝑦𝐵))
13 fpwrelmap.2 . . . . . . . 8 𝐵 ∈ V
1413ssex 5321 . . . . . . 7 ({𝑦𝑦 ∈ (𝑓𝑥)} ⊆ 𝐵 → {𝑦𝑦 ∈ (𝑓𝑥)} ∈ V)
1512, 14sylbir 235 . . . . . 6 (∀𝑦(𝑦 ∈ (𝑓𝑥) → 𝑦𝐵) → {𝑦𝑦 ∈ (𝑓𝑥)} ∈ V)
1611, 15syl 17 . . . . 5 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑥𝐴) → {𝑦𝑦 ∈ (𝑓𝑥)} ∈ V)
173, 16opabex3d 7990 . . . 4 (𝑓 ∈ (𝒫 𝐵m 𝐴) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ∈ V)
1817adantl 481 . . 3 ((⊤ ∧ 𝑓 ∈ (𝒫 𝐵m 𝐴)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ∈ V)
192mptex 7243 . . . 4 (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) ∈ V
2019a1i 11 . . 3 ((⊤ ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) → (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) ∈ V)
2110imdistanda 571 . . . . . . . . . 10 (𝑓 ∈ (𝒫 𝐵m 𝐴) → ((𝑥𝐴𝑦 ∈ (𝑓𝑥)) → (𝑥𝐴𝑦𝐵)))
2221ssopab2dv 5556 . . . . . . . . 9 (𝑓 ∈ (𝒫 𝐵m 𝐴) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)})
2322adantr 480 . . . . . . . 8 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)})
24 simpr 484 . . . . . . . 8 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
25 df-xp 5691 . . . . . . . . 9 (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)}
2625a1i 11 . . . . . . . 8 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)})
2723, 24, 263sstr4d 4039 . . . . . . 7 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑟 ⊆ (𝐴 × 𝐵))
28 velpw 4605 . . . . . . 7 (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↔ 𝑟 ⊆ (𝐴 × 𝐵))
2927, 28sylibr 234 . . . . . 6 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑟 ∈ 𝒫 (𝐴 × 𝐵))
305feqmptd 6977 . . . . . . . 8 (𝑓 ∈ (𝒫 𝐵m 𝐴) → 𝑓 = (𝑥𝐴 ↦ (𝑓𝑥)))
3130adantr 480 . . . . . . 7 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑓 = (𝑥𝐴 ↦ (𝑓𝑥)))
32 nfv 1914 . . . . . . . . 9 𝑥 𝑓 ∈ (𝒫 𝐵m 𝐴)
33 nfopab1 5213 . . . . . . . . . 10 𝑥{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}
3433nfeq2 2923 . . . . . . . . 9 𝑥 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}
3532, 34nfan 1899 . . . . . . . 8 𝑥(𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
36 df-rab 3437 . . . . . . . . . 10 {𝑦𝐵𝑥𝑟𝑦} = {𝑦 ∣ (𝑦𝐵𝑥𝑟𝑦)}
3736a1i 11 . . . . . . . . 9 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → {𝑦𝐵𝑥𝑟𝑦} = {𝑦 ∣ (𝑦𝐵𝑥𝑟𝑦)})
38 nfv 1914 . . . . . . . . . . . 12 𝑦 𝑓 ∈ (𝒫 𝐵m 𝐴)
39 nfopab2 5214 . . . . . . . . . . . . 13 𝑦{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}
4039nfeq2 2923 . . . . . . . . . . . 12 𝑦 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}
4138, 40nfan 1899 . . . . . . . . . . 11 𝑦(𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
42 nfv 1914 . . . . . . . . . . 11 𝑦 𝑥𝐴
4341, 42nfan 1899 . . . . . . . . . 10 𝑦((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴)
449adantllr 719 . . . . . . . . . . . . 13 ((((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑦𝐵)
45 df-br 5144 . . . . . . . . . . . . . . . . 17 (𝑥𝑟𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑟)
46 eleq2 2830 . . . . . . . . . . . . . . . . . 18 (𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} → (⟨𝑥, 𝑦⟩ ∈ 𝑟 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}))
47 opabidw 5529 . . . . . . . . . . . . . . . . . 18 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥)))
4846, 47bitrdi 287 . . . . . . . . . . . . . . . . 17 (𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} → (⟨𝑥, 𝑦⟩ ∈ 𝑟 ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
4945, 48bitrid 283 . . . . . . . . . . . . . . . 16 (𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} → (𝑥𝑟𝑦 ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
5049ad2antlr 727 . . . . . . . . . . . . . . 15 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑥𝑟𝑦 ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
51 elfvdm 6943 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝑓𝑥) → 𝑥 ∈ dom 𝑓)
5251adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑥 ∈ dom 𝑓)
535fdmd 6746 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ (𝒫 𝐵m 𝐴) → dom 𝑓 = 𝐴)
5453adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → dom 𝑓 = 𝐴)
5552, 54eleqtrd 2843 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑥𝐴)
5655ex 412 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ (𝒫 𝐵m 𝐴) → (𝑦 ∈ (𝑓𝑥) → 𝑥𝐴))
5756pm4.71rd 562 . . . . . . . . . . . . . . . 16 (𝑓 ∈ (𝒫 𝐵m 𝐴) → (𝑦 ∈ (𝑓𝑥) ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
5857ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑦 ∈ (𝑓𝑥) ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
5950, 58bitr4d 282 . . . . . . . . . . . . . 14 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑥𝑟𝑦𝑦 ∈ (𝑓𝑥)))
6059biimpar 477 . . . . . . . . . . . . 13 ((((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑥𝑟𝑦)
6144, 60jca 511 . . . . . . . . . . . 12 ((((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → (𝑦𝐵𝑥𝑟𝑦))
6261ex 412 . . . . . . . . . . 11 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑦 ∈ (𝑓𝑥) → (𝑦𝐵𝑥𝑟𝑦)))
6359biimpd 229 . . . . . . . . . . . 12 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑥𝑟𝑦𝑦 ∈ (𝑓𝑥)))
6463adantld 490 . . . . . . . . . . 11 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → ((𝑦𝐵𝑥𝑟𝑦) → 𝑦 ∈ (𝑓𝑥)))
6562, 64impbid 212 . . . . . . . . . 10 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑦 ∈ (𝑓𝑥) ↔ (𝑦𝐵𝑥𝑟𝑦)))
6643, 65abbid 2810 . . . . . . . . 9 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → {𝑦𝑦 ∈ (𝑓𝑥)} = {𝑦 ∣ (𝑦𝐵𝑥𝑟𝑦)})
67 abid2 2879 . . . . . . . . . 10 {𝑦𝑦 ∈ (𝑓𝑥)} = (𝑓𝑥)
6867a1i 11 . . . . . . . . 9 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → {𝑦𝑦 ∈ (𝑓𝑥)} = (𝑓𝑥))
6937, 66, 683eqtr2rd 2784 . . . . . . . 8 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑓𝑥) = {𝑦𝐵𝑥𝑟𝑦})
7035, 69mpteq2da 5240 . . . . . . 7 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → (𝑥𝐴 ↦ (𝑓𝑥)) = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
7131, 70eqtrd 2777 . . . . . 6 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
7229, 71jca 511 . . . . 5 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
73 ssrab2 4080 . . . . . . . . . . . 12 {𝑦𝐵𝑥𝑟𝑦} ⊆ 𝐵
7413, 73elpwi2 5335 . . . . . . . . . . 11 {𝑦𝐵𝑥𝑟𝑦} ∈ 𝒫 𝐵
7574a1i 11 . . . . . . . . . 10 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑥𝐴) → {𝑦𝐵𝑥𝑟𝑦} ∈ 𝒫 𝐵)
7675fmpttd 7135 . . . . . . . . 9 (𝑟 ∈ 𝒫 (𝐴 × 𝐵) → (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵)
7776adantr 480 . . . . . . . 8 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵)
78 simpr 484 . . . . . . . . 9 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
7978feq1d 6720 . . . . . . . 8 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑓:𝐴⟶𝒫 𝐵 ↔ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵))
8077, 79mpbird 257 . . . . . . 7 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑓:𝐴⟶𝒫 𝐵)
8113pwex 5380 . . . . . . . 8 𝒫 𝐵 ∈ V
8281, 2elmap 8911 . . . . . . 7 (𝑓 ∈ (𝒫 𝐵m 𝐴) ↔ 𝑓:𝐴⟶𝒫 𝐵)
8380, 82sylibr 234 . . . . . 6 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑓 ∈ (𝒫 𝐵m 𝐴))
84 elpwi 4607 . . . . . . . . . 10 (𝑟 ∈ 𝒫 (𝐴 × 𝐵) → 𝑟 ⊆ (𝐴 × 𝐵))
8584adantr 480 . . . . . . . . 9 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑟 ⊆ (𝐴 × 𝐵))
86 xpss 5701 . . . . . . . . 9 (𝐴 × 𝐵) ⊆ (V × V)
8785, 86sstrdi 3996 . . . . . . . 8 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑟 ⊆ (V × V))
88 df-rel 5692 . . . . . . . 8 (Rel 𝑟𝑟 ⊆ (V × V))
8987, 88sylibr 234 . . . . . . 7 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → Rel 𝑟)
90 relopabv 5831 . . . . . . . 8 Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}
9190a1i 11 . . . . . . 7 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
92 id 22 . . . . . . 7 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
93 nfv 1914 . . . . . . . . 9 𝑥 𝑟 ∈ 𝒫 (𝐴 × 𝐵)
94 nfmpt1 5250 . . . . . . . . . 10 𝑥(𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})
9594nfeq2 2923 . . . . . . . . 9 𝑥 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})
9693, 95nfan 1899 . . . . . . . 8 𝑥(𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
97 nfv 1914 . . . . . . . . 9 𝑦 𝑟 ∈ 𝒫 (𝐴 × 𝐵)
9842nfci 2893 . . . . . . . . . . 11 𝑦𝐴
99 nfrab1 3457 . . . . . . . . . . 11 𝑦{𝑦𝐵𝑥𝑟𝑦}
10098, 99nfmpt 5249 . . . . . . . . . 10 𝑦(𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})
101100nfeq2 2923 . . . . . . . . 9 𝑦 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})
10297, 101nfan 1899 . . . . . . . 8 𝑦(𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
103 nfcv 2905 . . . . . . . 8 𝑥𝑟
104 nfcv 2905 . . . . . . . 8 𝑦𝑟
105 brelg 32621 . . . . . . . . . . . . . . . 16 ((𝑟 ⊆ (𝐴 × 𝐵) ∧ 𝑥𝑟𝑦) → (𝑥𝐴𝑦𝐵))
10684, 105sylan 580 . . . . . . . . . . . . . . 15 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑥𝑟𝑦) → (𝑥𝐴𝑦𝐵))
107106adantlr 715 . . . . . . . . . . . . . 14 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → (𝑥𝐴𝑦𝐵))
108107simpld 494 . . . . . . . . . . . . 13 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑥𝐴)
109107simprd 495 . . . . . . . . . . . . . 14 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑦𝐵)
110 simpr 484 . . . . . . . . . . . . . 14 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑥𝑟𝑦)
11178fveq1d 6908 . . . . . . . . . . . . . . . . . 18 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑓𝑥) = ((𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})‘𝑥))
11213rabex 5339 . . . . . . . . . . . . . . . . . . 19 {𝑦𝐵𝑥𝑟𝑦} ∈ V
113 eqid 2737 . . . . . . . . . . . . . . . . . . . 20 (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})
114113fvmpt2 7027 . . . . . . . . . . . . . . . . . . 19 ((𝑥𝐴 ∧ {𝑦𝐵𝑥𝑟𝑦} ∈ V) → ((𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})‘𝑥) = {𝑦𝐵𝑥𝑟𝑦})
115112, 114mpan2 691 . . . . . . . . . . . . . . . . . 18 (𝑥𝐴 → ((𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})‘𝑥) = {𝑦𝐵𝑥𝑟𝑦})
116111, 115sylan9eq 2797 . . . . . . . . . . . . . . . . 17 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝐴) → (𝑓𝑥) = {𝑦𝐵𝑥𝑟𝑦})
117116eleq2d 2827 . . . . . . . . . . . . . . . 16 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝐴) → (𝑦 ∈ (𝑓𝑥) ↔ 𝑦 ∈ {𝑦𝐵𝑥𝑟𝑦}))
118 rabid 3458 . . . . . . . . . . . . . . . 16 (𝑦 ∈ {𝑦𝐵𝑥𝑟𝑦} ↔ (𝑦𝐵𝑥𝑟𝑦))
119117, 118bitrdi 287 . . . . . . . . . . . . . . 15 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝐴) → (𝑦 ∈ (𝑓𝑥) ↔ (𝑦𝐵𝑥𝑟𝑦)))
120108, 119syldan 591 . . . . . . . . . . . . . 14 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → (𝑦 ∈ (𝑓𝑥) ↔ (𝑦𝐵𝑥𝑟𝑦)))
121109, 110, 120mpbir2and 713 . . . . . . . . . . . . 13 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑦 ∈ (𝑓𝑥))
122108, 121jca 511 . . . . . . . . . . . 12 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → (𝑥𝐴𝑦 ∈ (𝑓𝑥)))
123122ex 412 . . . . . . . . . . 11 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑥𝑟𝑦 → (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
124119simplbda 499 . . . . . . . . . . . 12 ((((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑥𝑟𝑦)
125124expl 457 . . . . . . . . . . 11 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → ((𝑥𝐴𝑦 ∈ (𝑓𝑥)) → 𝑥𝑟𝑦))
126123, 125impbid 212 . . . . . . . . . 10 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑥𝑟𝑦 ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
12745, 126bitr3id 285 . . . . . . . . 9 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (⟨𝑥, 𝑦⟩ ∈ 𝑟 ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
128127, 47bitr4di 289 . . . . . . . 8 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (⟨𝑥, 𝑦⟩ ∈ 𝑟 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}))
12996, 102, 103, 104, 33, 39, 128eqrelrd2 32628 . . . . . . 7 (((Rel 𝑟 ∧ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))) → 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
13089, 91, 92, 129syl21anc 838 . . . . . 6 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
13183, 130jca 511 . . . . 5 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}))
13272, 131impbii 209 . . . 4 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ↔ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
133132a1i 11 . . 3 (⊤ → ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ↔ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))))
1341, 18, 20, 133f1od 7685 . 2 (⊤ → 𝑀:(𝒫 𝐵m 𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵))
135134mptru 1547 1 𝑀:(𝒫 𝐵m 𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538   = wceq 1540  wtru 1541  wcel 2108  {cab 2714  {crab 3436  Vcvv 3480  wss 3951  𝒫 cpw 4600  cop 4632   class class class wbr 5143  {copab 5205  cmpt 5225   × cxp 5683  dom cdm 5685  Rel wrel 5690  wf 6557  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  m cmap 8866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868
This theorem is referenced by:  fpwrelmapffs  32745
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