Theorem divsfval 17175

Description: Value of the function in qusval 17170. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.)

Hypotheses

Ref	Expression
ercpbl.r	⊢ (𝜑 → ∼ Er 𝑉)
ercpbl.v	⊢ (𝜑 → 𝑉 ∈ 𝑊)
ercpbl.f	⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )

Assertion

Ref	Expression
divsfval	⊢ (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ )

Distinct variable groups:   𝑥, ∼   𝑥,𝐴   𝑥,𝑉   𝜑,𝑥

Allowed substitution hints:   𝐹(𝑥)   𝑊(𝑥)

Proof of Theorem divsfval

Step	Hyp	Ref	Expression
1		ercpbl.v	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝑊)
2		ercpbl.r	. . . . . 6 ⊢ (𝜑 → ∼ Er 𝑉)
3	2	ecss 8502	. . . . 5 ⊢ (𝜑 → [𝐴] ∼ ⊆ 𝑉)
4	1, 3	ssexd 5243	. . . 4 ⊢ (𝜑 → [𝐴] ∼ ∈ V)
5		eceq1 8494	. . . . 5 ⊢ (𝑥 = 𝐴 → [𝑥] ∼ = [𝐴] ∼ )
6		ercpbl.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
7	5, 6	fvmptg 6855	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ [𝐴] ∼ ∈ V) → (𝐹‘𝐴) = [𝐴] ∼ )
8	4, 7	sylan2 592	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → (𝐹‘𝐴) = [𝐴] ∼ )
9	8	expcom 413	. 2 ⊢ (𝜑 → (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = [𝐴] ∼ ))
10	6	dmeqi 5802	. . . . . . . 8 ⊢ dom 𝐹 = dom (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
11	2	ecss 8502	. . . . . . . . . . 11 ⊢ (𝜑 → [𝑥] ∼ ⊆ 𝑉)
12	1, 11	ssexd 5243	. . . . . . . . . 10 ⊢ (𝜑 → [𝑥] ∼ ∈ V)
13	12	ralrimivw 3108	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 [𝑥] ∼ ∈ V)
14		dmmptg 6134	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝑉 [𝑥] ∼ ∈ V → dom (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) = 𝑉)
15	13, 14	syl 17	. . . . . . . 8 ⊢ (𝜑 → dom (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) = 𝑉)
16	10, 15	eqtrid 2790	. . . . . . 7 ⊢ (𝜑 → dom 𝐹 = 𝑉)
17	16	eleq2d 2824	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ 𝑉))
18	17	notbid 317	. . . . 5 ⊢ (𝜑 → (¬ 𝐴 ∈ dom 𝐹 ↔ ¬ 𝐴 ∈ 𝑉))
19		ndmfv 6786	. . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
20	18, 19	syl6bir 253	. . . 4 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝑉 → (𝐹‘𝐴) = ∅))
21		ecdmn0 8503	. . . . . 6 ⊢ (𝐴 ∈ dom ∼ ↔ [𝐴] ∼ ≠ ∅)
22		erdm 8466	. . . . . . . . 9 ⊢ ( ∼ Er 𝑉 → dom ∼ = 𝑉)
23	2, 22	syl 17	. . . . . . . 8 ⊢ (𝜑 → dom ∼ = 𝑉)
24	23	eleq2d 2824	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∈ dom ∼ ↔ 𝐴 ∈ 𝑉))
25	24	biimpd 228	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ dom ∼ → 𝐴 ∈ 𝑉))
26	21, 25	syl5bir 242	. . . . 5 ⊢ (𝜑 → ([𝐴] ∼ ≠ ∅ → 𝐴 ∈ 𝑉))
27	26	necon1bd 2960	. . . 4 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝑉 → [𝐴] ∼ = ∅))
28	20, 27	jcad 512	. . 3 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝑉 → ((𝐹‘𝐴) = ∅ ∧ [𝐴] ∼ = ∅)))
29		eqtr3 2764	. . 3 ⊢ (((𝐹‘𝐴) = ∅ ∧ [𝐴] ∼ = ∅) → (𝐹‘𝐴) = [𝐴] ∼ )
30	28, 29	syl6 35	. 2 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝑉 → (𝐹‘𝐴) = [𝐴] ∼ ))
31	9, 30	pm2.61d 179	1 ⊢ (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ )

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  Vcvv 3422  ∅c0 4253   ↦ cmpt 5153  dom cdm 5580  ‘cfv 6418   Er wer 8453  [cec 8454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fv 6426  df-er 8456  df-ec 8458

This theorem is referenced by:  ercpbllem  17176  qusaddvallem  17179  qusgrp2  18608  frgpmhm  19286  frgpup3lem  19298  qusring2  19774  qusrhm  20421