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Mirrors > Home > ILE Home > Th. List > en2d | GIF version |
Description: Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.) |
Ref | Expression |
---|---|
en2d.1 | ⊢ (𝜑 → 𝐴 ∈ V) |
en2d.2 | ⊢ (𝜑 → 𝐵 ∈ V) |
en2d.3 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ V)) |
en2d.4 | ⊢ (𝜑 → (𝑦 ∈ 𝐵 → 𝐷 ∈ V)) |
en2d.5 | ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷))) |
Ref | Expression |
---|---|
en2d | ⊢ (𝜑 → 𝐴 ≈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | en2d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) | |
2 | en2d.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ V) | |
3 | eqid 2177 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
4 | en2d.3 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ V)) | |
5 | 4 | imp 124 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ V) |
6 | en2d.4 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → 𝐷 ∈ V)) | |
7 | 6 | imp 124 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ V) |
8 | en2d.5 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷))) | |
9 | 3, 5, 7, 8 | f1od 6073 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1-onto→𝐵) |
10 | f1oen2g 6754 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) | |
11 | 1, 2, 9, 10 | syl3anc 1238 | 1 ⊢ (𝜑 → 𝐴 ≈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 Vcvv 2737 class class class wbr 4003 ↦ cmpt 4064 –1-1-onto→wf1o 5215 ≈ cen 6737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-en 6740 |
This theorem is referenced by: en2i 6769 map1 6811 |
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