Theorem dmqsblocks 39112

Description: If the pet 39110 span (𝑅 ⋉ (^◡ E ↾ 𝐴)) partitions 𝐴, then every block 𝑢 ∈ 𝐴 is of the form [𝑣] for some 𝑣 that not only lies in the domain but also has at least one internal element 𝑐 and at least one 𝑅-target 𝑏 (cf. also the comments of qseq 38907). It makes explicit that pet 39110 gives active representatives for each block, without ever forcing 𝑣 = 𝑢. (Contributed by Peter Mazsa, 23-Nov-2025.)

Assertion

Ref	Expression
dmqsblocks	⊢ ((dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴 → ∀𝑢 ∈ 𝐴 ∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))∃𝑏∃𝑐(𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏))

Distinct variable groups:   𝐴,𝑏,𝑐,𝑢,𝑣   𝑅,𝑏,𝑐,𝑢,𝑣

Proof of Theorem dmqsblocks

Step	Hyp	Ref	Expression
1		qseq 38907	. . 3 ⊢ ((dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴 ↔ ∀𝑢(𝑢 ∈ 𝐴 ↔ ∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴))))
2		eqab2 38446	. . 3 ⊢ (∀𝑢(𝑢 ∈ 𝐴 ↔ ∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴))) → ∀𝑢 ∈ 𝐴 ∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)))
3	1, 2	sylbi 217	. 2 ⊢ ((dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴 → ∀𝑢 ∈ 𝐴 ∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)))
4		rexanid 3085	. . . 4 ⊢ (∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))(𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴))) ↔ ∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)))
5		eldmxrncnvepres2 38620	. . . . . . . . . 10 ⊢ (𝑣 ∈ V → (𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ↔ (𝑣 ∈ 𝐴 ∧ ∃𝑐 𝑐 ∈ 𝑣 ∧ ∃𝑏 𝑣𝑅𝑏)))
6	5	elv 3445	. . . . . . . . 9 ⊢ (𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ↔ (𝑣 ∈ 𝐴 ∧ ∃𝑐 𝑐 ∈ 𝑣 ∧ ∃𝑏 𝑣𝑅𝑏))
7		3simpc 1150	. . . . . . . . 9 ⊢ ((𝑣 ∈ 𝐴 ∧ ∃𝑐 𝑐 ∈ 𝑣 ∧ ∃𝑏 𝑣𝑅𝑏) → (∃𝑐 𝑐 ∈ 𝑣 ∧ ∃𝑏 𝑣𝑅𝑏))
8	6, 7	sylbi 217	. . . . . . . 8 ⊢ (𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) → (∃𝑐 𝑐 ∈ 𝑣 ∧ ∃𝑏 𝑣𝑅𝑏))
9		exdistrv 1956	. . . . . . . . 9 ⊢ (∃𝑐∃𝑏(𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏) ↔ (∃𝑐 𝑐 ∈ 𝑣 ∧ ∃𝑏 𝑣𝑅𝑏))
10		excom 2167	. . . . . . . . 9 ⊢ (∃𝑐∃𝑏(𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏) ↔ ∃𝑏∃𝑐(𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏))
11	9, 10	bitr3i 277	. . . . . . . 8 ⊢ ((∃𝑐 𝑐 ∈ 𝑣 ∧ ∃𝑏 𝑣𝑅𝑏) ↔ ∃𝑏∃𝑐(𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏))
12	8, 11	sylib 218	. . . . . . 7 ⊢ (𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) → ∃𝑏∃𝑐(𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏))
13	12	anim1ci 616	. . . . . 6 ⊢ ((𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴))) → (𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ ∃𝑏∃𝑐(𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏)))
14		3anass 1094	. . . . . . . 8 ⊢ ((𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏) ↔ (𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏)))
15	14	2exbii 1850	. . . . . . 7 ⊢ (∃𝑏∃𝑐(𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏) ↔ ∃𝑏∃𝑐(𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏)))
16		19.42vv 1958	. . . . . . 7 ⊢ (∃𝑏∃𝑐(𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏)) ↔ (𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ ∃𝑏∃𝑐(𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏)))
17	15, 16	sylbbr 236	. . . . . 6 ⊢ ((𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ ∃𝑏∃𝑐(𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏)) → ∃𝑏∃𝑐(𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏))
18	13, 17	syl 17	. . . . 5 ⊢ ((𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴))) → ∃𝑏∃𝑐(𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏))
19	18	reximi 3074	. . . 4 ⊢ (∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))(𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴))) → ∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))∃𝑏∃𝑐(𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏))
20	4, 19	sylbir 235	. . 3 ⊢ (∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) → ∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))∃𝑏∃𝑐(𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏))
21	20	ralimi 3073	. 2 ⊢ (∀𝑢 ∈ 𝐴 ∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) → ∀𝑢 ∈ 𝐴 ∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))∃𝑏∃𝑐(𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏))
22	3, 21	syl 17	1 ⊢ ((dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴 → ∀𝑢 ∈ 𝐴 ∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))∃𝑏∃𝑐(𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060  Vcvv 3440   class class class wbr 5098   E cep 5523  ^◡ccnv 5623  dom cdm 5624   ↾ cres 5626  [cec 8633   / cqs 8634   ⋉ cxrn 38375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-eprel 5524  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fo 6498  df-fv 6500  df-oprab 7362  df-1st 7933  df-2nd 7934  df-qs 8641  df-xrn 38565

This theorem is referenced by: (None)