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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.) (Revised by AV, 4-Aug-2024.) | 
| Ref | Expression | 
|---|---|
| en2d.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) | 
| en2d.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) | 
| en2d.3 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝑋)) | 
| en2d.4 | ⊢ (𝜑 → (𝑦 ∈ 𝐵 → 𝐷 ∈ 𝑌)) | 
| 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 ∈ 𝑊) | |
| 3 | eqid 2737 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
| 4 | en2d.3 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝑋)) | |
| 5 | 4 | imp 406 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑋) | 
| 6 | en2d.4 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → 𝐷 ∈ 𝑌)) | |
| 7 | 6 | imp 406 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝑌) | 
| 8 | en2d.5 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷))) | |
| 9 | 3, 5, 7, 8 | f1od 7685 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1-onto→𝐵) | 
| 10 | f1oen2g 9009 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) | |
| 11 | 1, 2, 9, 10 | syl3anc 1373 | 1 ⊢ (𝜑 → 𝐴 ≈ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ↦ cmpt 5225 –1-1-onto→wf1o 6560 ≈ cen 8982 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-en 8986 | 
| This theorem is referenced by: en2i 9030 mapsnend 9076 snmapen 9078 gicsubgen 19297 lzenom 42781 | 
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