Theorem rabfodom 30045

Description: Domination relation for restricted abstract class builders, based on a surjective function. (Contributed by Thierry Arnoux, 27-Jan-2020.)

Hypotheses

Ref	Expression
rabfodom.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) → (𝜒 ↔ 𝜓))
rabfodom.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
rabfodom.3	⊢ (𝜑 → 𝐹:𝐴–onto→𝐵)

Assertion

Ref	Expression
rabfodom	⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝜒} ≼ {𝑥 ∈ 𝐴 ∣ 𝜓})

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦   𝑥,𝑉,𝑦   𝜑,𝑥,𝑦   𝜓,𝑦   𝜒,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)

Proof of Theorem rabfodom

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3418	. . . . . 6 ⊢ 𝑎 ∈ V
2	1	rabex 5091	. . . . 5 ⊢ {𝑥 ∈ 𝑎 ∣ 𝜓} ∈ V
3		eqid 2778	. . . . . 6 ⊢ (𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥))
4		rabfodom.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝐴–onto→𝐵)
5		fof 6419	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
6	4, 5	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
7	6	feqmptd 6562	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
8	7	ad2antrr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
9	8	reseq1d 5694	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → (𝐹 ↾ 𝑎) = ((𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) ↾ 𝑎))
10		elpwi 4432	. . . . . . . . . 10 ⊢ (𝑎 ∈ 𝒫 𝐴 → 𝑎 ⊆ 𝐴)
11	10	ad2antlr 714	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → 𝑎 ⊆ 𝐴)
12	11	resmptd 5753	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → ((𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) ↾ 𝑎) = (𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥)))
13	9, 12	eqtrd 2814	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → (𝐹 ↾ 𝑎) = (𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥)))
14		f1oeq1 6433	. . . . . . . 8 ⊢ ((𝐹 ↾ 𝑎) = (𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥)) → ((𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵 ↔ (𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥)):𝑎–1-1-onto→𝐵))
15	14	biimpa 469	. . . . . . 7 ⊢ (((𝐹 ↾ 𝑎) = (𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥)) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → (𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥)):𝑎–1-1-onto→𝐵)
16	13, 15	sylancom 579	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → (𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥)):𝑎–1-1-onto→𝐵)
17		simp1ll 1216	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 = (𝐹‘𝑥)) → 𝜑)
18	11	3ad2ant1 1113	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 = (𝐹‘𝑥)) → 𝑎 ⊆ 𝐴)
19		simp2 1117	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 = (𝐹‘𝑥)) → 𝑥 ∈ 𝑎)
20	18, 19	sseldd 3859	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 = (𝐹‘𝑥)) → 𝑥 ∈ 𝐴)
21		simp3 1118	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 = (𝐹‘𝑥)) → 𝑦 = (𝐹‘𝑥))
22		rabfodom.1	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) → (𝜒 ↔ 𝜓))
23	17, 20, 21, 22	syl3anc 1351	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 = (𝐹‘𝑥)) → (𝜒 ↔ 𝜓))
24	3, 16, 23	f1oresrab 6712	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → ((𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥)) ↾ {𝑥 ∈ 𝑎 ∣ 𝜓}):{𝑥 ∈ 𝑎 ∣ 𝜓}–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒})
25		f1oeng 8325	. . . . 5 ⊢ (({𝑥 ∈ 𝑎 ∣ 𝜓} ∈ V ∧ ((𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥)) ↾ {𝑥 ∈ 𝑎 ∣ 𝜓}):{𝑥 ∈ 𝑎 ∣ 𝜓}–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒}) → {𝑥 ∈ 𝑎 ∣ 𝜓} ≈ {𝑦 ∈ 𝐵 ∣ 𝜒})
26	2, 24, 25	sylancr 578	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → {𝑥 ∈ 𝑎 ∣ 𝜓} ≈ {𝑦 ∈ 𝐵 ∣ 𝜒})
27	26	ensymd 8357	. . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → {𝑦 ∈ 𝐵 ∣ 𝜒} ≈ {𝑥 ∈ 𝑎 ∣ 𝜓})
28		rabfodom.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
29		rabexg 5090	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
30	28, 29	syl 17	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
31	30	ad2antrr 713	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
32		rabss2 3944	. . . . 5 ⊢ (𝑎 ⊆ 𝐴 → {𝑥 ∈ 𝑎 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓})
33	11, 32	syl 17	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → {𝑥 ∈ 𝑎 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓})
34		ssdomg 8352	. . . 4 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V → ({𝑥 ∈ 𝑎 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓} → {𝑥 ∈ 𝑎 ∣ 𝜓} ≼ {𝑥 ∈ 𝐴 ∣ 𝜓}))
35	31, 33, 34	sylc 65	. . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → {𝑥 ∈ 𝑎 ∣ 𝜓} ≼ {𝑥 ∈ 𝐴 ∣ 𝜓})
36		endomtr 8364	. . 3 ⊢ (({𝑦 ∈ 𝐵 ∣ 𝜒} ≈ {𝑥 ∈ 𝑎 ∣ 𝜓} ∧ {𝑥 ∈ 𝑎 ∣ 𝜓} ≼ {𝑥 ∈ 𝐴 ∣ 𝜓}) → {𝑦 ∈ 𝐵 ∣ 𝜒} ≼ {𝑥 ∈ 𝐴 ∣ 𝜓})
37	27, 35, 36	syl2anc 576	. 2 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → {𝑦 ∈ 𝐵 ∣ 𝜒} ≼ {𝑥 ∈ 𝐴 ∣ 𝜓})
38		foresf1o 30044	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → ∃𝑎 ∈ 𝒫 𝐴(𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵)
39	28, 4, 38	syl2anc 576	. 2 ⊢ (𝜑 → ∃𝑎 ∈ 𝒫 𝐴(𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵)
40	37, 39	r19.29a 3234	1 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝜒} ≼ {𝑥 ∈ 𝐴 ∣ 𝜓})

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   ∧ w3a 1068   = wceq 1507   ∈ wcel 2050  ∃wrex 3089  {crab 3092  Vcvv 3415   ⊆ wss 3829  𝒫 cpw 4422   class class class wbr 4929   ↦ cmpt 5008   ↾ cres 5409  ⟶wf 6184  –onto→wfo 6186  –1-1-onto→wf1o 6187  ‘cfv 6188   ≈ cen 8303   ≼ cdom 8304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-reg 8851  ax-inf2 8898  ax-ac2 9683

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-iin 4795  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-se 5367  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-isom 6197  df-riota 6937  df-om 7397  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-er 8089  df-en 8307  df-dom 8308  df-r1 8987  df-rank 8988  df-card 9162  df-ac 9336

This theorem is referenced by:  locfinreflem  30754