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Mirrors > Home > NFE Home > Th. List > funeq | GIF version |
Description: Equality theorem for function predicate. (Contributed by set.mm contributors, 16-Aug-1994.) |
Ref | Expression |
---|---|
funeq | ⊢ (A = B → (Fun A ↔ Fun B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funss 5126 | . . . 4 ⊢ (B ⊆ A → (Fun A → Fun B)) | |
2 | funss 5126 | . . . 4 ⊢ (A ⊆ B → (Fun B → Fun A)) | |
3 | 1, 2 | anim12i 549 | . . 3 ⊢ ((B ⊆ A ∧ A ⊆ B) → ((Fun A → Fun B) ∧ (Fun B → Fun A))) |
4 | 3 | ancoms 439 | . 2 ⊢ ((A ⊆ B ∧ B ⊆ A) → ((Fun A → Fun B) ∧ (Fun B → Fun A))) |
5 | eqss 3287 | . 2 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A)) | |
6 | dfbi2 609 | . 2 ⊢ ((Fun A ↔ Fun B) ↔ ((Fun A → Fun B) ∧ (Fun B → Fun A))) | |
7 | 4, 5, 6 | 3imtr4i 257 | 1 ⊢ (A = B → (Fun A ↔ Fun B)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 = wceq 1642 ⊆ wss 3257 Fun wfun 4775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 df-opab 4623 df-br 4640 df-co 4726 df-cnv 4785 df-fun 4789 |
This theorem is referenced by: funeqi 5128 funeqd 5129 fununi 5160 funcnvuni 5161 cnvresid 5166 fneq1 5173 elfuns 5829 elfunsg 5830 elpmg 6013 fundmeng 6044 |
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