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Mirrors > Home > NFE Home > Th. List > rneqd | GIF version |
Description: Equality deduction for range. (Contributed by set.mm contributors, 4-Mar-2004.) |
Ref | Expression |
---|---|
rneqd.1 | ⊢ (φ → A = B) |
Ref | Expression |
---|---|
rneqd | ⊢ (φ → ran A = ran B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rneqd.1 | . 2 ⊢ (φ → A = B) | |
2 | rneq 4956 | . 2 ⊢ (A = B → ran A = ran B) | |
3 | 1, 2 | syl 15 | 1 ⊢ (φ → ran A = ran B) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 ran crn 4773 |
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-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-rex 2620 df-br 4640 df-ima 4727 df-rn 4786 |
This theorem is referenced by: resima2 5007 resiima 5012 rnxpid 5054 elxp4 5108 funimacnv 5168 fnima 5201 |
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