Theorem dmqsblocks 39616

Description: If the pet 39614 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 39382). It makes explicit that pet 39614 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 39382	. . 3 ⊢ ((dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴 ↔ ∀𝑢(𝑢 ∈ 𝐴 ↔ ∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴))))
2		eqab2 38899	. . 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 220	. 2 ⊢ ((dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴 → ∀𝑢 ∈ 𝐴 ∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)))
4		rexanid 3114	. . . 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 39084	. . . . . . . . . 10 ⊢ (𝑣 ∈ V → (𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ↔ (𝑣 ∈ 𝐴 ∧ ∃𝑐 𝑐 ∈ 𝑣 ∧ ∃𝑏 𝑣𝑅𝑏)))
6	5	elv 3460	. . . . . . . . 9 ⊢ (𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ↔ (𝑣 ∈ 𝐴 ∧ ∃𝑐 𝑐 ∈ 𝑣 ∧ ∃𝑏 𝑣𝑅𝑏))
7		3simpc 1168	. . . . . . . . 9 ⊢ ((𝑣 ∈ 𝐴 ∧ ∃𝑐 𝑐 ∈ 𝑣 ∧ ∃𝑏 𝑣𝑅𝑏) → (∃𝑐 𝑐 ∈ 𝑣 ∧ ∃𝑏 𝑣𝑅𝑏))
8	6, 7	sylbi 220	. . . . . . . 8 ⊢ (𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) → (∃𝑐 𝑐 ∈ 𝑣 ∧ ∃𝑏 𝑣𝑅𝑏))
9		exdistrv 1985	. . . . . . . . 9 ⊢ (∃𝑐∃𝑏(𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏) ↔ (∃𝑐 𝑐 ∈ 𝑣 ∧ ∃𝑏 𝑣𝑅𝑏))
10		excom 2197	. . . . . . . . 9 ⊢ (∃𝑐∃𝑏(𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏) ↔ ∃𝑏∃𝑐(𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏))
11	9, 10	bitr3i 280	. . . . . . . 8 ⊢ ((∃𝑐 𝑐 ∈ 𝑣 ∧ ∃𝑏 𝑣𝑅𝑏) ↔ ∃𝑏∃𝑐(𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏))
12	8, 11	sylib 221	. . . . . . 7 ⊢ (𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) → ∃𝑏∃𝑐(𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏))
13	12	anim1ci 627	. . . . . 6 ⊢ ((𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴))) → (𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ ∃𝑏∃𝑐(𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏)))
14		3anass 1111	. . . . . . . 8 ⊢ ((𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏) ↔ (𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏)))
15	14	2exbii 1879	. . . . . . 7 ⊢ (∃𝑏∃𝑐(𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏) ↔ ∃𝑏∃𝑐(𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏)))
16		19.42vv 1987	. . . . . . 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 239	. . . . . 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 18	. . . . 5 ⊢ ((𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴))) → ∃𝑏∃𝑐(𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏))
19	18	reximi 3103	. . . 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 238	. . 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 3102	. 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 18	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 209   ∧ wa 400   ∧ w3a 1103  ∀wal 1568   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  Vcvv 3455   class class class wbr 5109   E cep 5560  ^◡ccnv 5660  dom cdm 5661   ↾ cres 5663  [cec 8688   / cqs 8689   ⋉ cxrn 38823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-eprel 5561  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-oprab 7414  df-1st 7982  df-2nd 7983  df-qs 8696  df-xrn 39029

This theorem is referenced by: (None)