Theorem enssdom 8134

Description: Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.)

Assertion

Ref	Expression
enssdom	⊢ ≈ ⊆ ≼

Proof of Theorem enssdom

Dummy variables 𝑥 𝑦 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relen 8114	. 2 ⊢ Rel ≈
2		f1of1 6285	. . . . 5 ⊢ (𝑓:𝑥–1-1-onto→𝑦 → 𝑓:𝑥–1-1→𝑦)
3	2	eximi 1899	. . . 4 ⊢ (∃𝑓 𝑓:𝑥–1-1-onto→𝑦 → ∃𝑓 𝑓:𝑥–1-1→𝑦)
4		opabid 5120	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} ↔ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦)
5		opabid 5120	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦} ↔ ∃𝑓 𝑓:𝑥–1-1→𝑦)
6	3, 4, 5	3imtr4i 281	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} → ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦})
7		df-en 8110	. . . 4 ⊢ ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}
8	7	eleq2i 2819	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ≈ ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦})
9		df-dom 8111	. . . 4 ⊢ ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦}
10	9	eleq2i 2819	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ≼ ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦})
11	6, 8, 10	3imtr4i 281	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ≈ → ⟨𝑥, 𝑦⟩ ∈ ≼ )
12	1, 11	relssi 5356	1 ⊢ ≈ ⊆ ≼

Syntax hints:  ∃wex 1841   ∈ wcel 2127   ⊆ wss 3703  ⟨cop 4315  {copab 4852  –1-1→wf1 6034  –1-1-onto→wf1o 6036   ≈ cen 8106   ≼ cdom 8107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-rab 3047  df-v 3330  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-opab 4853  df-xp 5260  df-rel 5261  df-f1o 6044  df-en 8110  df-dom 8111

This theorem is referenced by:  dfdom2  8135  endom  8136