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 2482 | . 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 5486 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))) | |
6 | 3, 4, 5 | syl2anc 408 | . 2 ⊢ (𝜑 → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))) |
7 | 2, 6 | mpbird 166 | 1 ⊢ (𝜑 → 𝐹 = 𝐺) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1316 ∈ wcel 1465 ∀wral 2393 Fn wfn 5088 ‘cfv 5093 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-csb 2976 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fn 5096 df-fv 5101 |
This theorem is referenced by: foeqcnvco 5659 f1eqcocnv 5660 offeq 5963 tfrlem1 6173 frecrdg 6273 updjudhcoinlf 6933 updjudhcoinrg 6934 seq3val 10199 seqvalcd 10200 seq3feq2 10211 seq3feq 10213 seqfeq3 10253 seq3shft 10578 efcvgfsum 11300 upxp 12368 uptx 12370 dvidlemap 12756 dvrecap 12773 peano4nninf 13127 nninfalllemn 13129 nninfsellemeqinf 13139 nninffeq 13143 refeq 13150 |
Copyright terms: Public domain | W3C validator |