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Mirrors > Home > MPE Home > Th. List > foeq2 | Structured version Visualization version GIF version |
Description: Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.) |
Ref | Expression |
---|---|
foeq2 | ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–onto→𝐶 ↔ 𝐹:𝐵–onto→𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fneq2 6595 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐹 Fn 𝐴 ↔ 𝐹 Fn 𝐵)) | |
2 | 1 | anbi1d 631 | . 2 ⊢ (𝐴 = 𝐵 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐶) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 = 𝐶))) |
3 | df-fo 6503 | . 2 ⊢ (𝐹:𝐴–onto→𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐶)) | |
4 | df-fo 6503 | . 2 ⊢ (𝐹:𝐵–onto→𝐶 ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 = 𝐶)) | |
5 | 2, 3, 4 | 3bitr4g 314 | 1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–onto→𝐶 ↔ 𝐹:𝐵–onto→𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ran crn 5635 Fn wfn 6492 –onto→wfo 6495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-cleq 2729 df-fn 6500 df-fo 6503 |
This theorem is referenced by: foco 6771 f1oeq2 6774 foeq123d 6778 tposfo 8185 brwdom 9504 brwdom2 9510 canthwdom 9516 cfslb2n 10205 0ramcl 16896 ghmcyg 19674 txcmpb 22998 qtoptopon 23058 fsupprnfi 31610 opidon2OLD 36316 fnfocofob 45318 fullthinc 47073 |
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