Theorem dom2 6278

Description: A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. 𝐶 and 𝐷 can be read 𝐶(𝑥) and 𝐷(𝑦), as can be inferred from their distinct variable conditions. (Contributed by NM, 26-Oct-2003.)

Hypotheses

Ref	Expression
dom2.1	⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵)
dom2.2	⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦))

Assertion

Ref	Expression
dom2	⊢ (𝐵 ∈ 𝑉 → 𝐴 ≼ 𝐵)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑦,𝐶   𝑥,𝐷

Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem dom2

Step	Hyp	Ref	Expression
1		eqid 2081	. 2 ⊢ 𝐴 = 𝐴
2		dom2.1	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵)
3	2	a1i 9	. . 3 ⊢ (𝐴 = 𝐴 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵))
4		dom2.2	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦))
5	4	a1i 9	. . 3 ⊢ (𝐴 = 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦)))
6	3, 5	dom2d 6276	. 2 ⊢ (𝐴 = 𝐴 → (𝐵 ∈ 𝑉 → 𝐴 ≼ 𝐵))
7	1, 6	ax-mp 7	1 ⊢ (𝐵 ∈ 𝑉 → 𝐴 ≼ 𝐵)

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284   ∈ wcel 1433   class class class wbr 3785   ≼ cdom 6243

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-dom 6246

This theorem is referenced by: (None)