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Mirrors > Home > MPE Home > Th. List > fin1a2lem1 | Structured version Visualization version GIF version |
Description: Lemma for fin1a2 9837. (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 7528 | . 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) | |
2 | suceq 6256 | . . 3 ⊢ (𝑎 = 𝐴 → suc 𝑎 = suc 𝐴) | |
3 | fin1a2lem.a | . . . 4 ⊢ 𝑆 = (𝑥 ∈ On ↦ suc 𝑥) | |
4 | suceq 6256 | . . . . 5 ⊢ (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎) | |
5 | 4 | cbvmptv 5169 | . . . 4 ⊢ (𝑥 ∈ On ↦ suc 𝑥) = (𝑎 ∈ On ↦ suc 𝑎) |
6 | 3, 5 | eqtri 2844 | . . 3 ⊢ 𝑆 = (𝑎 ∈ On ↦ suc 𝑎) |
7 | 2, 6 | fvmptg 6766 | . 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 1537 ∈ wcel 2114 ↦ cmpt 5146 Oncon0 6191 suc csuc 6193 ‘cfv 6355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-ord 6194 df-on 6195 df-suc 6197 df-iota 6314 df-fun 6357 df-fv 6363 |
This theorem is referenced by: fin1a2lem2 9823 fin1a2lem6 9827 |
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