Theorem eldisjlem19 39508

Description: Special case of disjlem19 39499 (together with membpartlem19 39509, this is former prtlem19 39598). (Contributed by Peter Mazsa, 21-Oct-2021.)

Assertion

Ref	Expression
eldisjlem19	⊢ (𝐵 ∈ 𝑉 → ( ElDisj 𝐴 → ((𝑢 ∈ dom (◡ E ↾ 𝐴) ∧ 𝐵 ∈ 𝑢) → 𝑢 = [𝐵] ∼ 𝐴)))

Distinct variable groups:   𝑢,𝐴   𝑢,𝐵   𝑢,𝑉

Proof of Theorem eldisjlem19

Step	Hyp	Ref	Expression
1		df-eldisj 39387	. . . . . . . 8 ⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))
2		disjlem19 39499	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → ( Disj (◡ E ↾ 𝐴) → ((𝑢 ∈ dom (◡ E ↾ 𝐴) ∧ 𝐵 ∈ [𝑢](◡ E ↾ 𝐴)) → [𝑢](◡ E ↾ 𝐴) = [𝐵] ≀ (◡ E ↾ 𝐴))))
3	1, 2	biimtrid 245	. . . . . . 7 ⊢ (𝐵 ∈ 𝑉 → ( ElDisj 𝐴 → ((𝑢 ∈ dom (◡ E ↾ 𝐴) ∧ 𝐵 ∈ [𝑢](◡ E ↾ 𝐴)) → [𝑢](◡ E ↾ 𝐴) = [𝐵] ≀ (◡ E ↾ 𝐴))))
4	3	imp 411	. . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ ElDisj 𝐴) → ((𝑢 ∈ dom (◡ E ↾ 𝐴) ∧ 𝐵 ∈ [𝑢](◡ E ↾ 𝐴)) → [𝑢](◡ E ↾ 𝐴) = [𝐵] ≀ (◡ E ↾ 𝐴)))
5	4	expdimp 457	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ ElDisj 𝐴) ∧ 𝑢 ∈ dom (◡ E ↾ 𝐴)) → (𝐵 ∈ [𝑢](◡ E ↾ 𝐴) → [𝑢](◡ E ↾ 𝐴) = [𝐵] ≀ (◡ E ↾ 𝐴)))
6		eccnvepres3 38887	. . . . . . . 8 ⊢ (𝑢 ∈ dom (◡ E ↾ 𝐴) → [𝑢](◡ E ↾ 𝐴) = 𝑢)
7	6	eleq2d 2847	. . . . . . 7 ⊢ (𝑢 ∈ dom (◡ E ↾ 𝐴) → (𝐵 ∈ [𝑢](◡ E ↾ 𝐴) ↔ 𝐵 ∈ 𝑢))
8	6	eqeq1d 2763	. . . . . . 7 ⊢ (𝑢 ∈ dom (◡ E ↾ 𝐴) → ([𝑢](◡ E ↾ 𝐴) = [𝐵] ≀ (◡ E ↾ 𝐴) ↔ 𝑢 = [𝐵] ≀ (◡ E ↾ 𝐴)))
9	7, 8	imbi12d 347	. . . . . 6 ⊢ (𝑢 ∈ dom (◡ E ↾ 𝐴) → ((𝐵 ∈ [𝑢](◡ E ↾ 𝐴) → [𝑢](◡ E ↾ 𝐴) = [𝐵] ≀ (◡ E ↾ 𝐴)) ↔ (𝐵 ∈ 𝑢 → 𝑢 = [𝐵] ≀ (◡ E ↾ 𝐴))))
10	9	adantl 486	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ ElDisj 𝐴) ∧ 𝑢 ∈ dom (◡ E ↾ 𝐴)) → ((𝐵 ∈ [𝑢](◡ E ↾ 𝐴) → [𝑢](◡ E ↾ 𝐴) = [𝐵] ≀ (◡ E ↾ 𝐴)) ↔ (𝐵 ∈ 𝑢 → 𝑢 = [𝐵] ≀ (◡ E ↾ 𝐴))))
11	5, 10	mpbid 235	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ ElDisj 𝐴) ∧ 𝑢 ∈ dom (◡ E ↾ 𝐴)) → (𝐵 ∈ 𝑢 → 𝑢 = [𝐵] ≀ (◡ E ↾ 𝐴)))
12		df-coels 39097	. . . . . 6 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
13	12	eceq2i 8736	. . . . 5 ⊢ [𝐵] ∼ 𝐴 = [𝐵] ≀ (◡ E ↾ 𝐴)
14	13	eqeq2i 2774	. . . 4 ⊢ (𝑢 = [𝐵] ∼ 𝐴 ↔ 𝑢 = [𝐵] ≀ (◡ E ↾ 𝐴))
15	11, 14	imbitrrdi 255	. . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ ElDisj 𝐴) ∧ 𝑢 ∈ dom (◡ E ↾ 𝐴)) → (𝐵 ∈ 𝑢 → 𝑢 = [𝐵] ∼ 𝐴))
16	15	expimpd 458	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ ElDisj 𝐴) → ((𝑢 ∈ dom (◡ E ↾ 𝐴) ∧ 𝐵 ∈ 𝑢) → 𝑢 = [𝐵] ∼ 𝐴))
17	16	ex 417	1 ⊢ (𝐵 ∈ 𝑉 → ( ElDisj 𝐴 → ((𝑢 ∈ dom (◡ E ↾ 𝐴) ∧ 𝐵 ∈ 𝑢) → 𝑢 = [𝐵] ∼ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1568   ∈ wcel 2141   E cep 5560  ^◡ccnv 5660  dom cdm 5661   ↾ cres 5663  [cec 8691   ≀ ccoss 38778   ∼ ccoels 38779   Disj wdisjALTV 38814   ElDisj weldisj 38816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3367  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  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-ima 5674  df-ec 8695  df-coss 39096  df-coels 39097  df-cnvrefrel 39202  df-disjALTV 39385  df-eldisj 39387

This theorem is referenced by:  membpartlem19  39509