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Mirrors > Home > ILE Home > Th. List > funeq | GIF version |
Description: Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.) |
Ref | Expression |
---|---|
funeq | ⊢ (𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqimss2 3118 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴) | |
2 | funss 5100 | . . 3 ⊢ (𝐵 ⊆ 𝐴 → (Fun 𝐴 → Fun 𝐵)) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝐴 = 𝐵 → (Fun 𝐴 → Fun 𝐵)) |
4 | eqimss 3117 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) | |
5 | funss 5100 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (Fun 𝐵 → Fun 𝐴)) | |
6 | 4, 5 | syl 14 | . 2 ⊢ (𝐴 = 𝐵 → (Fun 𝐵 → Fun 𝐴)) |
7 | 3, 6 | impbid 128 | 1 ⊢ (𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1314 ⊆ wss 3037 Fun wfun 5075 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-in 3043 df-ss 3050 df-br 3896 df-opab 3950 df-rel 4506 df-cnv 4507 df-co 4508 df-fun 5083 |
This theorem is referenced by: funeqi 5102 funeqd 5103 fununi 5149 funcnvuni 5150 cnvresid 5155 fneq1 5169 elpmg 6512 fundmeng 6655 |
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