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Mirrors > Home > MPE Home > Th. List > Mathboxes > funALTVeq | Structured version Visualization version GIF version |
Description: Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.) |
Ref | Expression |
---|---|
funALTVeq | ⊢ (𝐴 = 𝐵 → ( FunALTV 𝐴 ↔ FunALTV 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqimss2 4005 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴) | |
2 | funALTVss 37211 | . . 3 ⊢ (𝐵 ⊆ 𝐴 → ( FunALTV 𝐴 → FunALTV 𝐵)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 = 𝐵 → ( FunALTV 𝐴 → FunALTV 𝐵)) |
4 | eqimss 4004 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) | |
5 | funALTVss 37211 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴)) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝐴 = 𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴)) |
7 | 3, 6 | impbid 211 | 1 ⊢ (𝐴 = 𝐵 → ( FunALTV 𝐴 ↔ FunALTV 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ⊆ wss 3914 FunALTV wfunALTV 36715 |
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-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-coss 36923 df-cnvrefrel 37039 df-funALTV 37194 |
This theorem is referenced by: funALTVeqi 37213 funALTVeqd 37214 |
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