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Theorem cnvf1olem 6279

Description: Lemma for cnvf1o 6280. (Contributed by Mario Carneiro, 27-Apr-2014.)

**Assertion**
Ref	Expression
cnvf1olem	⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → (𝐶 ∈ ^◡𝐴 ∧ 𝐵 = ∪ ^◡{𝐶}))

**Proof of Theorem cnvf1olem**
Step	Hyp	Ref	Expression
1		simprr 531	. . . 4 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐶 = ∪ ^◡{𝐵})
2		1st2nd 6236	. . . . . . . 8 ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
3	2	adantrr 479	. . . . . . 7 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
4	3	sneqd 3632	. . . . . 6 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → {𝐵} = {⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩})
5	4	cnveqd 4839	. . . . 5 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ^◡{𝐵} = ^◡{⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩})
6	5	unieqd 3847	. . . 4 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ∪ ^◡{𝐵} = ∪ ^◡{⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩})
7		1stexg 6222	. . . . . 6 ⊢ (𝐵 ∈ 𝐴 → (1^st ‘𝐵) ∈ V)
8		2ndexg 6223	. . . . . 6 ⊢ (𝐵 ∈ 𝐴 → (2^nd ‘𝐵) ∈ V)
9		opswapg 5153	. . . . . 6 ⊢ (((1^st ‘𝐵) ∈ V ∧ (2^nd ‘𝐵) ∈ V) → ∪ ^◡{⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩} = ⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩)
10	7, 8, 9	syl2anc 411	. . . . 5 ⊢ (𝐵 ∈ 𝐴 → ∪ ^◡{⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩} = ⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩)
11	10	ad2antrl 490	. . . 4 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ∪ ^◡{⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩} = ⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩)
12	1, 6, 11	3eqtrd 2230	. . 3 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐶 = ⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩)
13		simprl 529	. . . . 5 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐵 ∈ 𝐴)
14	3, 13	eqeltrrd 2271	. . . 4 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ 𝐴)
15		opelcnvg 4843	. . . . . 6 ⊢ (((2^nd ‘𝐵) ∈ V ∧ (1^st ‘𝐵) ∈ V) → (⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩ ∈ ^◡𝐴 ↔ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ 𝐴))
16	8, 7, 15	syl2anc 411	. . . . 5 ⊢ (𝐵 ∈ 𝐴 → (⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩ ∈ ^◡𝐴 ↔ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ 𝐴))
17	16	ad2antrl 490	. . . 4 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → (⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩ ∈ ^◡𝐴 ↔ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ 𝐴))
18	14, 17	mpbird 167	. . 3 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩ ∈ ^◡𝐴)
19	12, 18	eqeltrd 2270	. 2 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐶 ∈ ^◡𝐴)
20		opswapg 5153	. . . . . 6 ⊢ (((2^nd ‘𝐵) ∈ V ∧ (1^st ‘𝐵) ∈ V) → ∪ ^◡{⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩} = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
21	8, 7, 20	syl2anc 411	. . . . 5 ⊢ (𝐵 ∈ 𝐴 → ∪ ^◡{⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩} = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
22	21	eqcomd 2199	. . . 4 ⊢ (𝐵 ∈ 𝐴 → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ = ∪ ^◡{⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩})
23	22	ad2antrl 490	. . 3 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ = ∪ ^◡{⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩})
24	12	sneqd 3632	. . . . 5 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → {𝐶} = {⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩})
25	24	cnveqd 4839	. . . 4 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ^◡{𝐶} = ^◡{⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩})
26	25	unieqd 3847	. . 3 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ∪ ^◡{𝐶} = ∪ ^◡{⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩})
27	23, 3, 26	3eqtr4d 2236	. 2 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐵 = ∪ ^◡{𝐶})
28	19, 27	jca 306	1 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → (𝐶 ∈ ^◡𝐴 ∧ 𝐵 = ∪ ^◡{𝐶}))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2164 Vcvv 2760 {csn 3619 ⟨cop 3622 ∪ cuni 3836 ^◡ccnv 4659 Rel wrel 4665 ‘cfv 5255 1^st c1st 6193 2^nd c2nd 6194

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-sbc 2987 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-br 4031 df-opab 4092 df-mpt 4093 df-id 4325 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-fo 5261 df-fv 5263 df-1st 6195 df-2nd 6196

This theorem is referenced by: cnvf1o 6280

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