Theorem dom3d 6491

Description: A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by Mario Carneiro, 20-May-2013.)

Hypotheses

Ref	Expression
dom2d.1	⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵))
dom2d.2	⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦)))
dom3d.3	⊢ (𝜑 → 𝐴 ∈ 𝑉)
dom3d.4	⊢ (𝜑 → 𝐵 ∈ 𝑊)

Assertion

Ref	Expression
dom3d	⊢ (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)

Proof of Theorem dom3d

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dom2d.1	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵))
2		dom2d.2	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦)))
3	1, 2	dom2lem 6489	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1→𝐵)
4		f1f 5216	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1→𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶𝐵)
5	3, 4	syl 14	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶𝐵)
6		dom3d.3	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7		dom3d.4	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
8		fex2 5179	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ V)
9	5, 6, 7, 8	syl3anc 1174	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ V)
10		f1eq1 5211	. . . 4 ⊢ (𝑧 = (𝑥 ∈ 𝐴 ↦ 𝐶) → (𝑧:𝐴–1-1→𝐵 ↔ (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1→𝐵))
11	10	spcegv 2707	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ V → ((𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1→𝐵 → ∃𝑧 𝑧:𝐴–1-1→𝐵))
12	9, 3, 11	sylc 61	. 2 ⊢ (𝜑 → ∃𝑧 𝑧:𝐴–1-1→𝐵)
13		brdomg 6465	. . 3 ⊢ (𝐵 ∈ 𝑊 → (𝐴 ≼ 𝐵 ↔ ∃𝑧 𝑧:𝐴–1-1→𝐵))
14	7, 13	syl 14	. 2 ⊢ (𝜑 → (𝐴 ≼ 𝐵 ↔ ∃𝑧 𝑧:𝐴–1-1→𝐵))
15	12, 14	mpbird 165	1 ⊢ (𝜑 → 𝐴 ≼ 𝐵)

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1289  ∃wex 1426   ∈ wcel 1438  Vcvv 2619   class class class wbr 3845   ↦ cmpt 3899  ⟶wf 5011  –1-1→wf1 5012   ≼ cdom 6456

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fv 5023  df-dom 6459

This theorem is referenced by:  dom3  6493  xpdom2  6547  fopwdom  6552