Theorem rabfodom 32583

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 3445	. . . . . 6 ⊢ 𝑎 ∈ V
2	1	rabex 5285	. . . . 5 ⊢ {𝑥 ∈ 𝑎 ∣ 𝜓} ∈ V
3		eqid 2737	. . . . . 6 ⊢ (𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥))
4		rabfodom.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝐴–onto→𝐵)
5		fof 6747	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
6	4, 5	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
7	6	feqmptd 6903	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
8	7	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
9	8	reseq1d 5938	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → (𝐹 ↾ 𝑎) = ((𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) ↾ 𝑎))
10		elpwi 4562	. . . . . . . . . 10 ⊢ (𝑎 ∈ 𝒫 𝐴 → 𝑎 ⊆ 𝐴)
11	10	ad2antlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → 𝑎 ⊆ 𝐴)
12	11	resmptd 6000	. . . . . . . 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 2772	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → (𝐹 ↾ 𝑎) = (𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥)))
14		f1oeq1 6763	. . . . . . . 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 589	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → (𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥)):𝑎–1-1-onto→𝐵)
17		simp1ll 1238	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 = (𝐹‘𝑥)) → 𝜑)
18	11	3ad2ant1 1134	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 = (𝐹‘𝑥)) → 𝑎 ⊆ 𝐴)
19		simp2 1138	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 = (𝐹‘𝑥)) → 𝑥 ∈ 𝑎)
20	18, 19	sseldd 3935	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 = (𝐹‘𝑥)) → 𝑥 ∈ 𝐴)
21		simp3 1139	. . . . . . 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 1374	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 = (𝐹‘𝑥)) → (𝜒 ↔ 𝜓))
24	3, 16, 23	f1oresrab 7074	. . . . 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 8911	. . . . 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 588	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → {𝑥 ∈ 𝑎 ∣ 𝜓} ≈ {𝑦 ∈ 𝐵 ∣ 𝜒})
27	26	ensymd 8946	. . 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 5283	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
30	28, 29	syl 17	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
31	30	ad2antrr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
32		rabss2 4030	. . . . 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 8941	. . . 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 8953	. . 3 ⊢ (({𝑦 ∈ 𝐵 ∣ 𝜒} ≈ {𝑥 ∈ 𝑎 ∣ 𝜓} ∧ {𝑥 ∈ 𝑎 ∣ 𝜓} ≼ {𝑥 ∈ 𝐴 ∣ 𝜓}) → {𝑦 ∈ 𝐵 ∣ 𝜒} ≼ {𝑥 ∈ 𝐴 ∣ 𝜓})
37	27, 35, 36	syl2anc 585	. 2 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → {𝑦 ∈ 𝐵 ∣ 𝜒} ≼ {𝑥 ∈ 𝐴 ∣ 𝜓})
38		foresf1o 32582	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → ∃𝑎 ∈ 𝒫 𝐴(𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵)
39	28, 4, 38	syl2anc 585	. 2 ⊢ (𝜑 → ∃𝑎 ∈ 𝒫 𝐴(𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵)
40	37, 39	r19.29a 3145	1 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝜒} ≼ {𝑥 ∈ 𝐴 ∣ 𝜓})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3061  {crab 3400  Vcvv 3441   ⊆ wss 3902  𝒫 cpw 4555   class class class wbr 5099   ↦ cmpt 5180   ↾ cres 5627  ⟶wf 6489  –onto→wfo 6491  –1-1-onto→wf1o 6492  ‘cfv 6493   ≈ cen 8884   ≼ cdom 8885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-reg 9501  ax-inf2 9554  ax-ac2 10377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-iin 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7317  df-ov 7363  df-om 7811  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-er 8637  df-en 8888  df-dom 8889  df-r1 9680  df-rank 9681  df-card 9855  df-ac 10030

This theorem is referenced by:  locfinreflem  33999