Theorem en3d 6942

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 2231	. . 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 6228	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1-onto→𝐵)
11		f1oen2g 6928	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)
12	1, 2, 10, 11	syl3anc 1273	1 ⊢ (𝜑 → 𝐴 ≈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1397   ∈ wcel 2202  Vcvv 2802   class class class wbr 4088   ↦ cmpt 4150  –1-1-onto→wf1o 5325   ≈ cen 6907

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-en 6910

This theorem is referenced by:  en3i  6944  fundmen  6981  mapen  7032  mapxpen  7034  ssenen  7037  fzen  10278  uzennn  10699  hashfacen  11101  hashdvds  12795