Theorem divsfval 17509

Description: Value of the function in qusval 17504. (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 8692	. . . . 5 ⊢ (𝜑 → [𝐴] ∼ ⊆ 𝑉)
4	1, 3	ssexd 5259	. . . 4 ⊢ (𝜑 → [𝐴] ∼ ∈ V)
5		eceq1 8680	. . . . 5 ⊢ (𝑥 = 𝐴 → [𝑥] ∼ = [𝐴] ∼ )
6		ercpbl.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
7	5, 6	fvmptg 6940	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ [𝐴] ∼ ∈ V) → (𝐹‘𝐴) = [𝐴] ∼ )
8	4, 7	sylan2 599	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → (𝐹‘𝐴) = [𝐴] ∼ )
9	8	expcom 414	. 2 ⊢ (𝜑 → (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = [𝐴] ∼ ))
10	6	dmeqi 5853	. . . . . . . 8 ⊢ dom 𝐹 = dom (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
11	2	ecss 8692	. . . . . . . . . . 11 ⊢ (𝜑 → [𝑥] ∼ ⊆ 𝑉)
12	1, 11	ssexd 5259	. . . . . . . . . 10 ⊢ (𝜑 → [𝑥] ∼ ∈ V)
13	12	ralrimivw 3136	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 [𝑥] ∼ ∈ V)
14		dmmptg 6200	. . . . . . . . 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 2787	. . . . . . 7 ⊢ (𝜑 → dom 𝐹 = 𝑉)
17	16	eleq2d 2826	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ 𝑉))
18	17	notbid 319	. . . . 5 ⊢ (𝜑 → (¬ 𝐴 ∈ dom 𝐹 ↔ ¬ 𝐴 ∈ 𝑉))
19		ndmfv 6866	. . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
20	18, 19	biimtrrdi 255	. . . 4 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝑉 → (𝐹‘𝐴) = ∅))
21		ecdmn0 8693	. . . . . 6 ⊢ (𝐴 ∈ dom ∼ ↔ [𝐴] ∼ ≠ ∅)
22		erdm 8651	. . . . . . . . 9 ⊢ ( ∼ Er 𝑉 → dom ∼ = 𝑉)
23	2, 22	syl 17	. . . . . . . 8 ⊢ (𝜑 → dom ∼ = 𝑉)
24	23	eleq2d 2826	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∈ dom ∼ ↔ 𝐴 ∈ 𝑉))
25	24	biimpd 230	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ dom ∼ → 𝐴 ∈ 𝑉))
26	21, 25	biimtrrid 244	. . . . 5 ⊢ (𝜑 → ([𝐴] ∼ ≠ ∅ → 𝐴 ∈ 𝑉))
27	26	necon1bd 2953	. . . 4 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝑉 → [𝐴] ∼ = ∅))
28	20, 27	jcad 517	. . 3 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝑉 → ((𝐹‘𝐴) = ∅ ∧ [𝐴] ∼ = ∅)))
29		eqtr3 2762	. . 3 ⊢ (((𝐹‘𝐴) = ∅ ∧ [𝐴] ∼ = ∅) → (𝐹‘𝐴) = [𝐴] ∼ )
30	28, 29	syl6 35	. 2 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝑉 → (𝐹‘𝐴) = [𝐴] ∼ ))
31	9, 30	pm2.61d 180	1 ⊢ (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ )

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  ∀wral 3054  Vcvv 3432  ∅c0 4268   ↦ cmpt 5160  dom cdm 5625  ‘cfv 6492   Er wer 8637  [cec 8638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  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-iota 6448  df-fun 6494  df-fv 6500  df-er 8640  df-ec 8642

This theorem is referenced by:  ercpbllem  17510  qusaddvallem  17513  qusgrp2  19032  frgpmhm  19738  frgpup3lem  19750  qusring2  20312  qusrhm  21276