Theorem brwdom3 9048

Description: Condition for weak dominance with a condition reminiscent of wdomd 9047. (Contributed by Stefan O'Rear, 13-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)

Assertion

Ref	Expression
brwdom3	⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝑋 ≼* 𝑌 ↔ ∃𝑓∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦)))

Distinct variable groups:   𝑓,𝑋,𝑥,𝑦   𝑓,𝑌,𝑥,𝑦

Allowed substitution hints:   𝑉(𝑥,𝑦,𝑓)   𝑊(𝑥,𝑦,𝑓)

Proof of Theorem brwdom3

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3514	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
2		elex 3514	. 2 ⊢ (𝑌 ∈ 𝑊 → 𝑌 ∈ V)
3		brwdom2 9039	. . . . 5 ⊢ (𝑌 ∈ V → (𝑋 ≼* 𝑌 ↔ ∃𝑧 ∈ 𝒫 𝑌∃𝑓 𝑓:𝑧–onto→𝑋))
4	3	adantl 484	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 ≼* 𝑌 ↔ ∃𝑧 ∈ 𝒫 𝑌∃𝑓 𝑓:𝑧–onto→𝑋))
5		dffo3 6870	. . . . . . . 8 ⊢ (𝑓:𝑧–onto→𝑋 ↔ (𝑓:𝑧⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑧 𝑥 = (𝑓‘𝑦)))
6	5	simprbi 499	. . . . . . 7 ⊢ (𝑓:𝑧–onto→𝑋 → ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑧 𝑥 = (𝑓‘𝑦))
7		elpwi 4550	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝒫 𝑌 → 𝑧 ⊆ 𝑌)
8		ssrexv 4036	. . . . . . . . . 10 ⊢ (𝑧 ⊆ 𝑌 → (∃𝑦 ∈ 𝑧 𝑥 = (𝑓‘𝑦) → ∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦)))
9	7, 8	syl 17	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝒫 𝑌 → (∃𝑦 ∈ 𝑧 𝑥 = (𝑓‘𝑦) → ∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦)))
10	9	adantl 484	. . . . . . . 8 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑧 ∈ 𝒫 𝑌) → (∃𝑦 ∈ 𝑧 𝑥 = (𝑓‘𝑦) → ∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦)))
11	10	ralimdv 3180	. . . . . . 7 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑧 ∈ 𝒫 𝑌) → (∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑧 𝑥 = (𝑓‘𝑦) → ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦)))
12	6, 11	syl5 34	. . . . . 6 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑧 ∈ 𝒫 𝑌) → (𝑓:𝑧–onto→𝑋 → ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦)))
13	12	eximdv 1918	. . . . 5 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑧 ∈ 𝒫 𝑌) → (∃𝑓 𝑓:𝑧–onto→𝑋 → ∃𝑓∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦)))
14	13	rexlimdva 3286	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (∃𝑧 ∈ 𝒫 𝑌∃𝑓 𝑓:𝑧–onto→𝑋 → ∃𝑓∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦)))
15	4, 14	sylbid 242	. . 3 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 ≼* 𝑌 → ∃𝑓∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦)))
16		simpll 765	. . . . . 6 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦)) → 𝑋 ∈ V)
17		simplr 767	. . . . . 6 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦)) → 𝑌 ∈ V)
18		eqeq1 2827	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑥 = (𝑓‘𝑦) ↔ 𝑧 = (𝑓‘𝑦)))
19	18	rexbidv 3299	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦) ↔ ∃𝑦 ∈ 𝑌 𝑧 = (𝑓‘𝑦)))
20		fveq2 6672	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → (𝑓‘𝑦) = (𝑓‘𝑤))
21	20	eqeq2d 2834	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (𝑧 = (𝑓‘𝑦) ↔ 𝑧 = (𝑓‘𝑤)))
22	21	cbvrexvw 3452	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝑌 𝑧 = (𝑓‘𝑦) ↔ ∃𝑤 ∈ 𝑌 𝑧 = (𝑓‘𝑤))
23	19, 22	syl6bb 289	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦) ↔ ∃𝑤 ∈ 𝑌 𝑧 = (𝑓‘𝑤)))
24	23	cbvralvw 3451	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦) ↔ ∀𝑧 ∈ 𝑋 ∃𝑤 ∈ 𝑌 𝑧 = (𝑓‘𝑤))
25	24	biimpi 218	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦) → ∀𝑧 ∈ 𝑋 ∃𝑤 ∈ 𝑌 𝑧 = (𝑓‘𝑤))
26	25	adantl 484	. . . . . . 7 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦)) → ∀𝑧 ∈ 𝑋 ∃𝑤 ∈ 𝑌 𝑧 = (𝑓‘𝑤))
27	26	r19.21bi 3210	. . . . . 6 ⊢ ((((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦)) ∧ 𝑧 ∈ 𝑋) → ∃𝑤 ∈ 𝑌 𝑧 = (𝑓‘𝑤))
28	16, 17, 27	wdom2d 9046	. . . . 5 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦)) → 𝑋 ≼* 𝑌)
29	28	ex 415	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦) → 𝑋 ≼* 𝑌))
30	29	exlimdv 1934	. . 3 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (∃𝑓∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦) → 𝑋 ≼* 𝑌))
31	15, 30	impbid 214	. 2 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 ≼* 𝑌 ↔ ∃𝑓∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦)))
32	1, 2, 31	syl2an 597	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝑋 ≼* 𝑌 ↔ ∃𝑓∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141  Vcvv 3496   ⊆ wss 3938  𝒫 cpw 4541   class class class wbr 5068  ⟶wf 6353  –onto→wfo 6355  ‘cfv 6357   ≼^* cwdom 9023

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-wdom 9025

This theorem is referenced by:  brwdom3i  9049