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Mirrors > Home > MPE Home > Th. List > fin1a2lem2 | Structured version Visualization version GIF version |
Description: Lemma for fin1a2 9839. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
Ref | Expression |
---|---|
fin1a2lem.a | ⊢ 𝑆 = (𝑥 ∈ On ↦ suc 𝑥) |
Ref | Expression |
---|---|
fin1a2lem2 | ⊢ 𝑆:On–1-1→On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fin1a2lem.a | . . 3 ⊢ 𝑆 = (𝑥 ∈ On ↦ suc 𝑥) | |
2 | suceloni 7530 | . . 3 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On) | |
3 | 1, 2 | fmpti 6878 | . 2 ⊢ 𝑆:On⟶On |
4 | 1 | fin1a2lem1 9824 | . . . . . 6 ⊢ (𝑎 ∈ On → (𝑆‘𝑎) = suc 𝑎) |
5 | 1 | fin1a2lem1 9824 | . . . . . 6 ⊢ (𝑏 ∈ On → (𝑆‘𝑏) = suc 𝑏) |
6 | 4, 5 | eqeqan12d 2840 | . . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑆‘𝑎) = (𝑆‘𝑏) ↔ suc 𝑎 = suc 𝑏)) |
7 | suc11 6296 | . . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (suc 𝑎 = suc 𝑏 ↔ 𝑎 = 𝑏)) | |
8 | 6, 7 | bitrd 281 | . . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑆‘𝑎) = (𝑆‘𝑏) ↔ 𝑎 = 𝑏)) |
9 | 8 | biimpd 231 | . . 3 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑆‘𝑎) = (𝑆‘𝑏) → 𝑎 = 𝑏)) |
10 | 9 | rgen2 3205 | . 2 ⊢ ∀𝑎 ∈ On ∀𝑏 ∈ On ((𝑆‘𝑎) = (𝑆‘𝑏) → 𝑎 = 𝑏) |
11 | dff13 7015 | . 2 ⊢ (𝑆:On–1-1→On ↔ (𝑆:On⟶On ∧ ∀𝑎 ∈ On ∀𝑏 ∈ On ((𝑆‘𝑎) = (𝑆‘𝑏) → 𝑎 = 𝑏))) | |
12 | 3, 10, 11 | mpbir2an 709 | 1 ⊢ 𝑆:On–1-1→On |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ↦ cmpt 5148 Oncon0 6193 suc csuc 6195 ⟶wf 6353 –1-1→wf1 6354 ‘cfv 6357 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-ord 6196 df-on 6197 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fv 6365 |
This theorem is referenced by: fin1a2lem6 9829 |
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