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Mirrors > Home > MPE Home > Th. List > f1eq3 | Structured version Visualization version GIF version |
Description: Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.) |
Ref | Expression |
---|---|
f1eq3 | ⊢ (𝐴 = 𝐵 → (𝐹:𝐶–1-1→𝐴 ↔ 𝐹:𝐶–1-1→𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | feq3 6470 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶⟶𝐴 ↔ 𝐹:𝐶⟶𝐵)) | |
2 | 1 | anbi1d 632 | . 2 ⊢ (𝐴 = 𝐵 → ((𝐹:𝐶⟶𝐴 ∧ Fun ◡𝐹) ↔ (𝐹:𝐶⟶𝐵 ∧ Fun ◡𝐹))) |
3 | df-f1 6329 | . 2 ⊢ (𝐹:𝐶–1-1→𝐴 ↔ (𝐹:𝐶⟶𝐴 ∧ Fun ◡𝐹)) | |
4 | df-f1 6329 | . 2 ⊢ (𝐹:𝐶–1-1→𝐵 ↔ (𝐹:𝐶⟶𝐵 ∧ Fun ◡𝐹)) | |
5 | 2, 3, 4 | 3bitr4g 317 | 1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶–1-1→𝐴 ↔ 𝐹:𝐶–1-1→𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ◡ccnv 5518 Fun wfun 6318 ⟶wf 6320 –1-1→wf1 6321 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-in 3888 df-ss 3898 df-f 6328 df-f1 6329 |
This theorem is referenced by: f1oeq3 6581 f1eq123d 6583 tposf12 7900 brdomg 8502 pwfseq 10075 f1linds 20514 isusgrs 26949 usgrstrrepe 27025 usgrexilem 27230 tocycval 30800 diaf1oN 38426 |
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