Theorem en3d 6768

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
en3d.1	⊢ (𝜑 → 𝐴 ∈ V)
en3d.2	⊢ (𝜑 → 𝐵 ∈ V)
en3d.3	⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵))
en3d.4	⊢ (𝜑 → (𝑦 ∈ 𝐵 → 𝐷 ∈ 𝐴))
en3d.5	⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶)))

Assertion

Ref	Expression
en3d	⊢ (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 en3d

Step	Hyp	Ref	Expression
1		en3d.1	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
2		en3d.2	. 2 ⊢ (𝜑 → 𝐵 ∈ V)
3		eqid 2177	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)
4		en3d.3	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵))
5	4	imp 124	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
6		en3d.4	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → 𝐷 ∈ 𝐴))
7	6	imp 124	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝐴)
8		en3d.5	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶)))
9	8	imp 124	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶))
10	3, 5, 7, 9	f1o2d 6075	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1-onto→𝐵)
11		f1oen2g 6754	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)
12	1, 2, 10, 11	syl3anc 1238	1 ⊢ (𝜑 → 𝐴 ≈ 𝐵)

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:  en3i  6770  fundmen  6805  mapen  6845  mapxpen  6847  ssenen  6850  fzen  10040  uzennn  10433  hashfacen  10811  hashdvds  12215