Theorem dom2 9002

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 2760	. 2 ⊢ 𝐴 = 𝐴
2		dom2.1	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵)
3	2	a1i 11	. . 3 ⊢ (𝐴 = 𝐴 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵))
4		dom2.2	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦))
5	4	a1i 11	. . 3 ⊢ (𝐴 = 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦)))
6	3, 5	dom2d 9000	. 2 ⊢ (𝐴 = 𝐴 → (𝐵 ∈ 𝑉 → 𝐴 ≼ 𝐵))
7	1, 6	ax-mp 5	1 ⊢ (𝐵 ∈ 𝑉 → 𝐴 ≼ 𝐵)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   class class class wbr 5103   ≼ cdom 8951

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 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7737

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  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 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-dom 8955

This theorem is used by:  infpwfidom  10032  rpnnen1lem6  13033  rpnnen2lem12  16314  tgdom  23204  vitali  25842  rpnnen3  43874