Theorem en2d 6613

Description: Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.)

Hypotheses

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)))

Assertion

Ref	Expression
en2d	⊢ (𝜑 → 𝐴 ≈ 𝐵)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑦,𝐶   𝑥,𝐷   𝜑,𝑥,𝑦

Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑦)

Proof of Theorem en2d

Step	Hyp	Ref	Expression
1		en2d.1	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
2		en2d.2	. 2 ⊢ (𝜑 → 𝐵 ∈ V)
3		eqid 2113	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)
4		en2d.3	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ V))
5	4	imp 123	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ V)
6		en2d.4	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → 𝐷 ∈ V))
7	6	imp 123	. . 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 5925	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1-onto→𝐵)
10		f1oen2g 6600	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)
11	1, 2, 9, 10	syl3anc 1197	1 ⊢ (𝜑 → 𝐴 ≈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1312   ∈ wcel 1461  Vcvv 2655   class class class wbr 3893   ↦ cmpt 3947  –1-1-onto→wf1o 5078   ≈ cen 6583

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-en 6586

This theorem is referenced by:  en2i  6615  map1  6657