Theorem dom3d 9054

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 9052	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1→𝐵)
4		f1f 6817	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1→𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶𝐵)
5	3, 4	syl 17	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶𝐵)
6		dom3d.3	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7		dom3d.4	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
8		fex2 7974	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ V)
9	5, 6, 7, 8	syl3anc 1371	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ V)
10		f1eq1 6812	. . 3 ⊢ (𝑧 = (𝑥 ∈ 𝐴 ↦ 𝐶) → (𝑧:𝐴–1-1→𝐵 ↔ (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1→𝐵))
11	9, 3, 10	spcedv 3611	. 2 ⊢ (𝜑 → ∃𝑧 𝑧:𝐴–1-1→𝐵)
12		brdomg 9016	. . 3 ⊢ (𝐵 ∈ 𝑊 → (𝐴 ≼ 𝐵 ↔ ∃𝑧 𝑧:𝐴–1-1→𝐵))
13	7, 12	syl 17	. 2 ⊢ (𝜑 → (𝐴 ≼ 𝐵 ↔ ∃𝑧 𝑧:𝐴–1-1→𝐵))
14	11, 13	mpbird 257	1 ⊢ (𝜑 → 𝐴 ≼ 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  Vcvv 3488   class class class wbr 5166   ↦ cmpt 5249  ⟶wf 6569  –1-1→wf1 6570   ≼ cdom 9001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fv 6581  df-dom 9005

This theorem is referenced by:  dom3  9056  xpdom2  9133  fopwdom  9146