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Mirrors > Home > MPE Home > Th. List > fin1a2lem1 | Structured version Visualization version GIF version |
Description: Lemma for fin1a2 10309. (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 7738 | . 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) | |
2 | suceq 6381 | . . 3 ⊢ (𝑎 = 𝐴 → suc 𝑎 = suc 𝐴) | |
3 | fin1a2lem.a | . . . 4 ⊢ 𝑆 = (𝑥 ∈ On ↦ suc 𝑥) | |
4 | suceq 6381 | . . . . 5 ⊢ (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎) | |
5 | 4 | cbvmptv 5216 | . . . 4 ⊢ (𝑥 ∈ On ↦ suc 𝑥) = (𝑎 ∈ On ↦ suc 𝑎) |
6 | 3, 5 | eqtri 2764 | . . 3 ⊢ 𝑆 = (𝑎 ∈ On ↦ suc 𝑎) |
7 | 2, 6 | fvmptg 6943 | . 2 ⊢ ((𝐴 ∈ On ∧ suc 𝐴 ∈ On) → (𝑆‘𝐴) = suc 𝐴) |
8 | 1, 7 | mpdan 685 | 1 ⊢ (𝐴 ∈ On → (𝑆‘𝐴) = suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ↦ cmpt 5186 Oncon0 6315 suc csuc 6317 ‘cfv 6493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-ord 6318 df-on 6319 df-suc 6321 df-iota 6445 df-fun 6495 df-fv 6501 |
This theorem is referenced by: fin1a2lem2 10295 fin1a2lem6 10299 onsucf1o 41510 |
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