Theorem dom2d 9013

Description: A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by NM, 24-Jul-2004.) (Revised 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)))

Assertion

Ref	Expression
dom2d	⊢ (𝜑 → (𝐵 ∈ 𝑅 → 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)   𝑅(𝑥, 𝑦)

Proof of Theorem dom2d

Step	Hyp	Ref	Expression
1		dom2d.1	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵))
2		dom2d.2	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦)))
3	1, 2	dom2lem 9012	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1→𝐵)
4		f1domg 8991	. 2 ⊢ (𝐵 ∈ 𝑅 → ((𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1→𝐵 → 𝐴 ≼ 𝐵))
5	3, 4	syl5com 32	1 ⊢ (𝜑 → (𝐵 ∈ 𝑅 → 𝐴 ≼ 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   class class class wbr 5103   ↦ cmpt 5186  –1-1→wf1 6534   ≼ cdom 8964

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-dom 8968

This theorem is used by:  dom2  9015  fineqvlem  9250  fseqdom  10098  fin1a2lem9  10479  iundom2g  10617  canthwe  10729  prmreclem2  17088  prmreclem3  17089  sylow1lem4  19808  aannenlem1  26648  derangenlem  35915  fphpd  43802  pellexlem3  43817  unxpwdom3  44081