f1eq1 - Metamath Proof Explorer

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Theorem f1eq1 6009

Description: Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)

**Assertion**
Ref	Expression
f1eq1	⊢ (𝐹 = 𝐺 → (𝐹:𝐴–1-1→𝐵 ↔ 𝐺:𝐴–1-1→𝐵))

**Proof of Theorem f1eq1**
Step	Hyp	Ref	Expression
1		feq1 5939	. . 3 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐴⟶𝐵))
2		cnveq 5218	. . . 4 ⊢ (𝐹 = 𝐺 → ^◡𝐹 = ^◡𝐺)
3	2	funeqd 5825	. . 3 ⊢ (𝐹 = 𝐺 → (Fun ^◡𝐹 ↔ Fun ^◡𝐺))
4	1, 3	anbi12d 743	. 2 ⊢ (𝐹 = 𝐺 → ((𝐹:𝐴⟶𝐵 ∧ Fun ^◡𝐹) ↔ (𝐺:𝐴⟶𝐵 ∧ Fun ^◡𝐺)))
5		df-f1 5809	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ^◡𝐹))
6		df-f1 5809	. 2 ⊢ (𝐺:𝐴–1-1→𝐵 ↔ (𝐺:𝐴⟶𝐵 ∧ Fun ^◡𝐺))
7	4, 5, 6	3bitr4g 302	1 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴–1-1→𝐵 ↔ 𝐺:𝐴–1-1→𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ^◡ccnv 5037 Fun wfun 5798 ⟶wf 5800 –1-1→wf1 5801

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809

This theorem is referenced by: f1oeq1 6040 f1eq123d 6044 fo00 6084 f1prex 6439 fun11iun 7019 tposf12 7264 oacomf1olem 7531 f1dom2g 7859 f1domg 7861 dom3d 7883 domtr 7895 domssex2 8005 1sdom 8048 marypha1lem 8222 fseqenlem1 8730 dfac12lem2 8849 dfac12lem3 8850 ackbij2 8948 fin23lem28 9045 fin23lem32 9049 fin23lem34 9051 fin23lem35 9052 fin23lem41 9057 iundom2g 9241 pwfseqlem5 9364 hashf1lem1 13096 hashf1lem2 13097 hashf1 13098 4sqlem11 15497 conjsubgen 17516 sylow1lem2 17837 sylow2blem1 17858 hauspwpwf1 21601 istrkg2ld 25159 axlowdim 25641 isuslgra 25872 isusgra 25873 usgrares 25898 sizeusglecusg 26014 2trllemE 26083 constr1trl 26118 specval 28141 aciunf1lem 28844 zrhchr 29348 qqhre 29392 eldioph2lem2 36342 meadjiunlem 39358 sizusglecusg 40679