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Mirrors > Home > NFE Home > Th. List > tfineq | GIF version |
Description: Equality theorem for the finite T operator. (Contributed by SF, 24-Jan-2015.) |
Ref | Expression |
---|---|
tfineq | ⊢ (A = B → Tfin A = Tfin B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2359 | . . 3 ⊢ (A = B → (A = ∅ ↔ B = ∅)) | |
2 | rexeq 2808 | . . . . 5 ⊢ (A = B → (∃y ∈ A ℘1y ∈ x ↔ ∃y ∈ B ℘1y ∈ x)) | |
3 | 2 | anbi2d 684 | . . . 4 ⊢ (A = B → ((x ∈ Nn ∧ ∃y ∈ A ℘1y ∈ x) ↔ (x ∈ Nn ∧ ∃y ∈ B ℘1y ∈ x))) |
4 | 3 | iotabidv 4360 | . . 3 ⊢ (A = B → (℩x(x ∈ Nn ∧ ∃y ∈ A ℘1y ∈ x)) = (℩x(x ∈ Nn ∧ ∃y ∈ B ℘1y ∈ x))) |
5 | 1, 4 | ifbieq2d 3682 | . 2 ⊢ (A = B → if(A = ∅, ∅, (℩x(x ∈ Nn ∧ ∃y ∈ A ℘1y ∈ x))) = if(B = ∅, ∅, (℩x(x ∈ Nn ∧ ∃y ∈ B ℘1y ∈ x)))) |
6 | df-tfin 4443 | . 2 ⊢ Tfin A = if(A = ∅, ∅, (℩x(x ∈ Nn ∧ ∃y ∈ A ℘1y ∈ x))) | |
7 | df-tfin 4443 | . 2 ⊢ Tfin B = if(B = ∅, ∅, (℩x(x ∈ Nn ∧ ∃y ∈ B ℘1y ∈ x))) | |
8 | 5, 6, 7 | 3eqtr4g 2410 | 1 ⊢ (A = B → Tfin A = Tfin B) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 = wceq 1642 ∈ wcel 1710 ∃wrex 2615 ∅c0 3550 ifcif 3662 ℘1cpw1 4135 ℩cio 4337 Nn cnnc 4373 Tfin ctfin 4435 |
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-rex 2620 df-rab 2623 df-v 2861 df-nin 3211 df-compl 3212 df-un 3214 df-if 3663 df-uni 3892 df-iota 4339 df-tfin 4443 |
This theorem is referenced by: tfincl 4492 tfin11 4493 tfin1c 4499 tfinltfinlem1 4500 tfinltfin 4501 eventfin 4517 oddtfin 4518 sfintfin 4532 tfinnn 4534 vfinncvntnn 4548 vfinspsslem1 4550 vfinspss 4551 vfinspclt 4552 |
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