Theorem rabfodom 32706

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 3460	. . . . . 6 ⊢ 𝑎 ∈ V
2	1	rabex 5297	. . . . 5 ⊢ {𝑥 ∈ 𝑎 ∣ 𝜓} ∈ V
3		eqid 2764	. . . . . 6 ⊢ (𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥))
4		rabfodom.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝐴–onto→𝐵)
5		fof 6780	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
6	4, 5	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
7	6	feqmptd 6937	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
8	7	ad2antrr 736	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
9	8	reseq1d 5966	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → (𝐹 ↾ 𝑎) = ((𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) ↾ 𝑎))
10		elpwi 4564	. . . . . . . . . 10 ⊢ (𝑎 ∈ 𝒫 𝐴 → 𝑎 ⊆ 𝐴)
11	10	ad2antlr 737	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → 𝑎 ⊆ 𝐴)
12	11	resmptd 6031	. . . . . . . 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 2799	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → (𝐹 ↾ 𝑎) = (𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥)))
14		f1oeq1 6796	. . . . . . . 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 480	. . . . . . 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 597	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → (𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥)):𝑎–1-1-onto→𝐵)
17		simp1ll 1251	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 = (𝐹‘𝑥)) → 𝜑)
18	11	3ad2ant1 1147	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 = (𝐹‘𝑥)) → 𝑎 ⊆ 𝐴)
19		simp2 1151	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 = (𝐹‘𝑥)) → 𝑥 ∈ 𝑎)
20	18, 19	sseldd 3939	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 = (𝐹‘𝑥)) → 𝑥 ∈ 𝐴)
21		simp3 1152	. . . . . . 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 1392	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 = (𝐹‘𝑥)) → (𝜒 ↔ 𝜓))
24	3, 16, 23	f1oresrab 7111	. . . . 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 8953	. . . . 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 596	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → {𝑥 ∈ 𝑎 ∣ 𝜓} ≈ {𝑦 ∈ 𝐵 ∣ 𝜒})
27	26	ensymd 8988	. . 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 5295	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
30	28, 29	syl 17	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
31	30	ad2antrr 736	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
32		rabss2 4032	. . . . 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 8983	. . . 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 8995	. . 3 ⊢ (({𝑦 ∈ 𝐵 ∣ 𝜒} ≈ {𝑥 ∈ 𝑎 ∣ 𝜓} ∧ {𝑥 ∈ 𝑎 ∣ 𝜓} ≼ {𝑥 ∈ 𝐴 ∣ 𝜓}) → {𝑦 ∈ 𝐵 ∣ 𝜒} ≼ {𝑥 ∈ 𝐴 ∣ 𝜓})
37	27, 35, 36	syl2anc 593	. 2 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → {𝑦 ∈ 𝐵 ∣ 𝜒} ≼ {𝑥 ∈ 𝐴 ∣ 𝜓})
38		foresf1o 32705	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → ∃𝑎 ∈ 𝒫 𝐴(𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵)
39	28, 4, 38	syl2anc 593	. 2 ⊢ (𝜑 → ∃𝑎 ∈ 𝒫 𝐴(𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵)
40	37, 39	r19.29a 3172	1 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝜒} ≼ {𝑥 ∈ 𝐴 ∣ 𝜓})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  ∃wrex 3088  {crab 3416  Vcvv 3456   ⊆ wss 3906  𝒫 cpw 4557   class class class wbr 5102   ↦ cmpt 5183   ↾ cres 5651  ⟶wf 6519  –onto→wfo 6521  –1-1-onto→wf1o 6522  ‘cfv 6523   ≈ cen 8926   ≼ cdom 8927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-reg 9542  ax-inf2 9598  ax-ac2 10422

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-se 5603  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-isom 6532  df-riota 7355  df-ov 7401  df-om 7849  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-er 8680  df-en 8930  df-dom 8931  df-r1 9724  df-rank 9725  df-card 9899  df-ac 10074

This theorem is referenced by:  locfinreflem  34139