Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > eqfnfvd | GIF version |
Description: Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014.) |
Ref | Expression |
---|---|
eqfnfvd.1 | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
eqfnfvd.2 | ⊢ (𝜑 → 𝐺 Fn 𝐴) |
eqfnfvd.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥)) |
Ref | Expression |
---|---|
eqfnfvd | ⊢ (𝜑 → 𝐹 = 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqfnfvd.3 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥)) | |
2 | 1 | ralrimiva 2434 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)) |
3 | eqfnfvd.1 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
4 | eqfnfvd.2 | . . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴) | |
5 | eqfnfv 5286 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))) | |
6 | 3, 4, 5 | syl2anc 403 | . 2 ⊢ (𝜑 → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))) |
7 | 2, 6 | mpbird 165 | 1 ⊢ (𝜑 → 𝐹 = 𝐺) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1284 ∈ wcel 1433 ∀wral 2348 Fn wfn 4917 ‘cfv 4922 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-csb 2909 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fn 4925 df-fv 4930 |
This theorem is referenced by: foeqcnvco 5450 f1eqcocnv 5451 tfrlem1 5946 frecrdg 6015 iseqss 9446 iseqfeq2 9449 iseqfeq 9451 |
Copyright terms: Public domain | W3C validator |