Theorem rabfodom 32533

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 3482	. . . . . 6 ⊢ 𝑎 ∈ V
2	1	rabex 5345	. . . . 5 ⊢ {𝑥 ∈ 𝑎 ∣ 𝜓} ∈ V
3		eqid 2735	. . . . . 6 ⊢ (𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥))
4		rabfodom.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝐴–onto→𝐵)
5		fof 6821	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
6	4, 5	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
7	6	feqmptd 6977	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
8	7	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
9	8	reseq1d 5999	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → (𝐹 ↾ 𝑎) = ((𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) ↾ 𝑎))
10		elpwi 4612	. . . . . . . . . 10 ⊢ (𝑎 ∈ 𝒫 𝐴 → 𝑎 ⊆ 𝐴)
11	10	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → 𝑎 ⊆ 𝐴)
12	11	resmptd 6060	. . . . . . . 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 2775	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → (𝐹 ↾ 𝑎) = (𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥)))
14		f1oeq1 6837	. . . . . . . 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 476	. . . . . . 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 588	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → (𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥)):𝑎–1-1-onto→𝐵)
17		simp1ll 1235	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 = (𝐹‘𝑥)) → 𝜑)
18	11	3ad2ant1 1132	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 = (𝐹‘𝑥)) → 𝑎 ⊆ 𝐴)
19		simp2 1136	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 = (𝐹‘𝑥)) → 𝑥 ∈ 𝑎)
20	18, 19	sseldd 3996	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 = (𝐹‘𝑥)) → 𝑥 ∈ 𝐴)
21		simp3 1137	. . . . . . 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 1370	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 = (𝐹‘𝑥)) → (𝜒 ↔ 𝜓))
24	3, 16, 23	f1oresrab 7147	. . . . 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 9010	. . . . 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 587	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → {𝑥 ∈ 𝑎 ∣ 𝜓} ≈ {𝑦 ∈ 𝐵 ∣ 𝜒})
27	26	ensymd 9044	. . 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 5343	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
30	28, 29	syl 17	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
31	30	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
32		rabss2 4088	. . . . 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 9039	. . . 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 9051	. . 3 ⊢ (({𝑦 ∈ 𝐵 ∣ 𝜒} ≈ {𝑥 ∈ 𝑎 ∣ 𝜓} ∧ {𝑥 ∈ 𝑎 ∣ 𝜓} ≼ {𝑥 ∈ 𝐴 ∣ 𝜓}) → {𝑦 ∈ 𝐵 ∣ 𝜒} ≼ {𝑥 ∈ 𝐴 ∣ 𝜓})
37	27, 35, 36	syl2anc 584	. 2 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → {𝑦 ∈ 𝐵 ∣ 𝜒} ≼ {𝑥 ∈ 𝐴 ∣ 𝜓})
38		foresf1o 32532	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → ∃𝑎 ∈ 𝒫 𝐴(𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵)
39	28, 4, 38	syl2anc 584	. 2 ⊢ (𝜑 → ∃𝑎 ∈ 𝒫 𝐴(𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵)
40	37, 39	r19.29a 3160	1 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝜒} ≼ {𝑥 ∈ 𝐴 ∣ 𝜓})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∃wrex 3068  {crab 3433  Vcvv 3478   ⊆ wss 3963  𝒫 cpw 4605   class class class wbr 5148   ↦ cmpt 5231   ↾ cres 5691  ⟶wf 6559  –onto→wfo 6561  –1-1-onto→wf1o 6562  ‘cfv 6563   ≈ cen 8981   ≼ cdom 8982

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf2 9679  ax-ac2 10501

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-r1 9802  df-rank 9803  df-card 9977  df-ac 10154

This theorem is referenced by:  locfinreflem  33801