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Mirrors > Home > MPE Home > Th. List > funeq | Structured version Visualization version 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 3621 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴) | |
2 | funss 5822 | . . 3 ⊢ (𝐵 ⊆ 𝐴 → (Fun 𝐴 → Fun 𝐵)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 = 𝐵 → (Fun 𝐴 → Fun 𝐵)) |
4 | eqimss 3620 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) | |
5 | funss 5822 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (Fun 𝐵 → Fun 𝐴)) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝐴 = 𝐵 → (Fun 𝐵 → Fun 𝐴)) |
7 | 3, 6 | impbid 201 | 1 ⊢ (𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ⊆ wss 3540 Fun wfun 5798 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-in 3547 df-ss 3554 df-br 4584 df-opab 4644 df-rel 5045 df-cnv 5046 df-co 5047 df-fun 5806 |
This theorem is referenced by: funeqi 5824 funeqd 5825 fununi 5878 cnvresid 5882 fneq1 5893 funop 6320 funsndifnop 6321 nvof1o 6436 funcnvuni 7012 elpmg 7759 fundmeng 7917 isfsupp 8162 dfac9 8841 axdc3lem2 9156 frlmphllem 19938 structvtxvallem 25697 locfinreflem 29235 orvcval 29846 bnj1379 30155 bnj1385 30157 bnj1497 30382 elfunsg 31193 funop1 40327 usgredgop 40400 |
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