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Mirrors > Home > MPE Home > Th. List > fin1a2lem2 | Structured version Visualization version GIF version |
Description: Lemma for fin1a2 9120. (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 6905 | . . 3 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On) | |
3 | 1, 2 | fmpti 6291 | . 2 ⊢ 𝑆:On⟶On |
4 | 1 | fin1a2lem1 9105 | . . . . . 6 ⊢ (𝑎 ∈ On → (𝑆‘𝑎) = suc 𝑎) |
5 | 1 | fin1a2lem1 9105 | . . . . . 6 ⊢ (𝑏 ∈ On → (𝑆‘𝑏) = suc 𝑏) |
6 | 4, 5 | eqeqan12d 2626 | . . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑆‘𝑎) = (𝑆‘𝑏) ↔ suc 𝑎 = suc 𝑏)) |
7 | suc11 5748 | . . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (suc 𝑎 = suc 𝑏 ↔ 𝑎 = 𝑏)) | |
8 | 6, 7 | bitrd 267 | . . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑆‘𝑎) = (𝑆‘𝑏) ↔ 𝑎 = 𝑏)) |
9 | 8 | biimpd 218 | . . 3 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑆‘𝑎) = (𝑆‘𝑏) → 𝑎 = 𝑏)) |
10 | 9 | rgen2a 2960 | . 2 ⊢ ∀𝑎 ∈ On ∀𝑏 ∈ On ((𝑆‘𝑎) = (𝑆‘𝑏) → 𝑎 = 𝑏) |
11 | dff13 6416 | . 2 ⊢ (𝑆:On–1-1→On ↔ (𝑆:On⟶On ∧ ∀𝑎 ∈ On ∀𝑏 ∈ On ((𝑆‘𝑎) = (𝑆‘𝑏) → 𝑎 = 𝑏))) | |
12 | 3, 10, 11 | mpbir2an 957 | 1 ⊢ 𝑆:On–1-1→On |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ↦ cmpt 4643 Oncon0 5640 suc csuc 5642 ⟶wf 5800 –1-1→wf1 5801 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fv 5812 |
This theorem is referenced by: fin1a2lem6 9110 |
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