Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > fin1a2lem1 | Structured version Visualization version GIF version |
Description: Lemma for fin1a2 10012. (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 | suceloni 7581 | . 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) | |
2 | suceq 6267 | . . 3 ⊢ (𝑎 = 𝐴 → suc 𝑎 = suc 𝐴) | |
3 | fin1a2lem.a | . . . 4 ⊢ 𝑆 = (𝑥 ∈ On ↦ suc 𝑥) | |
4 | suceq 6267 | . . . . 5 ⊢ (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎) | |
5 | 4 | cbvmptv 5147 | . . . 4 ⊢ (𝑥 ∈ On ↦ suc 𝑥) = (𝑎 ∈ On ↦ suc 𝑎) |
6 | 3, 5 | eqtri 2762 | . . 3 ⊢ 𝑆 = (𝑎 ∈ On ↦ suc 𝑎) |
7 | 2, 6 | fvmptg 6805 | . 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 1543 ∈ wcel 2110 ↦ cmpt 5124 Oncon0 6202 suc csuc 6204 ‘cfv 6369 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-ord 6205 df-on 6206 df-suc 6208 df-iota 6327 df-fun 6371 df-fv 6377 |
This theorem is referenced by: fin1a2lem2 9998 fin1a2lem6 10002 |
Copyright terms: Public domain | W3C validator |