Theorem en2d 8998

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.)

Hypotheses

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)))

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)   𝑉(𝑥, 𝑦)   𝑊(𝑥, 𝑦)   𝑋(𝑥, 𝑦)   𝑌(𝑥, 𝑦)

Proof of Theorem en2d

Step	Hyp	Ref	Expression
1		en2d.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		en2d.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
3		eqid 2762	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)
4		en2d.3	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝑋))
5	4	imp 412	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑋)
6		en2d.4	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → 𝐷 ∈ 𝑌))
7	6	imp 412	. . 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 7670	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1-onto→𝐵)
10		f1oen2g 8978	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)
11	1, 2, 9, 10	syl3anc 1398	1 ⊢ (𝜑 → 𝐴 ≈ 𝐵)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   class class class wbr 5107   ↦ cmpt 5190  –1-1-onto→wf1o 6536   ≈ cen 8953

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-en 8957

This theorem is used by:  en2i  9000  mapsnend  9047  snmapen  9049  gicsubgen  19412  lzenom  43623