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Mirrors > Home > MPE Home > Th. List > en2d | Structured version Visualization version 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 2823 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
4 | en2d.3 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ V)) | |
5 | 4 | imp 409 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ V) |
6 | en2d.4 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → 𝐷 ∈ V)) | |
7 | 6 | imp 409 | . . 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 7399 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1-onto→𝐵) |
10 | f1oen2g 8528 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) | |
11 | 1, 2, 9, 10 | syl3anc 1367 | 1 ⊢ (𝜑 → 𝐴 ≈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 class class class wbr 5068 ↦ cmpt 5148 –1-1-onto→wf1o 6356 ≈ cen 8508 |
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-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 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-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-en 8512 |
This theorem is referenced by: en2i 8549 mapsnend 8590 snmapen 8592 gicsubgen 18420 lzenom 39374 |
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