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Mirrors > Home > MPE Home > Th. List > fin1a2lem1 | Structured version Visualization version GIF version |
Description: Lemma for fin1a2 10452. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
Ref | Expression |
---|---|
fin1a2lem.a | ⊢ 𝑆 = (𝑥 ∈ On ↦ suc 𝑥) |
Ref | Expression |
---|---|
fin1a2lem1 | ⊢ (𝐴 ∈ On → (𝑆‘𝐴) = suc 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onsuc 7830 | . 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) | |
2 | suceq 6451 | . . 3 ⊢ (𝑎 = 𝐴 → suc 𝑎 = suc 𝐴) | |
3 | fin1a2lem.a | . . . 4 ⊢ 𝑆 = (𝑥 ∈ On ↦ suc 𝑥) | |
4 | suceq 6451 | . . . . 5 ⊢ (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎) | |
5 | 4 | cbvmptv 5260 | . . . 4 ⊢ (𝑥 ∈ On ↦ suc 𝑥) = (𝑎 ∈ On ↦ suc 𝑎) |
6 | 3, 5 | eqtri 2762 | . . 3 ⊢ 𝑆 = (𝑎 ∈ On ↦ suc 𝑎) |
7 | 2, 6 | fvmptg 7013 | . 2 ⊢ ((𝐴 ∈ On ∧ suc 𝐴 ∈ On) → (𝑆‘𝐴) = suc 𝐴) |
8 | 1, 7 | mpdan 687 | 1 ⊢ (𝐴 ∈ On → (𝑆‘𝐴) = suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 ↦ cmpt 5230 Oncon0 6385 suc csuc 6387 ‘cfv 6562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-ord 6388 df-on 6389 df-suc 6391 df-iota 6515 df-fun 6564 df-fv 6570 |
This theorem is referenced by: fin1a2lem2 10438 fin1a2lem6 10442 onsucf1o 43261 |
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