Theorem dom3d 6752

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 6750	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1→𝐵)
4		f1f 5403	. . . . 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 5366	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ V)
9	5, 6, 7, 8	syl3anc 1233	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ V)
10		f1eq1 5398	. . . 4 ⊢ (𝑧 = (𝑥 ∈ 𝐴 ↦ 𝐶) → (𝑧:𝐴–1-1→𝐵 ↔ (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1→𝐵))
11	10	spcegv 2818	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ V → ((𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1→𝐵 → ∃𝑧 𝑧:𝐴–1-1→𝐵))
12	9, 3, 11	sylc 62	. 2 ⊢ (𝜑 → ∃𝑧 𝑧:𝐴–1-1→𝐵)
13		brdomg 6726	. . 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 166	1 ⊢ (𝜑 → 𝐴 ≼ 𝐵)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348  ∃wex 1485   ∈ wcel 2141  Vcvv 2730   class class class wbr 3989   ↦ cmpt 4050  ⟶wf 5194  –1-1→wf1 5195   ≼ cdom 6717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fv 5206  df-dom 6720

This theorem is referenced by:  dom3  6754  xpdom2  6809  fopwdom  6814