Theorem eldisjlem19 39083

Description: Special case of disjlem19 39074 (together with membpartlem19 39084, this is former prtlem19 39173). (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 38962	. . . . . . . 8 ⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))
2		disjlem19 39074	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → ( Disj (◡ E ↾ 𝐴) → ((𝑢 ∈ dom (◡ E ↾ 𝐴) ∧ 𝐵 ∈ [𝑢](◡ E ↾ 𝐴)) → [𝑢](◡ E ↾ 𝐴) = [𝐵] ≀ (◡ E ↾ 𝐴))))
3	1, 2	biimtrid 242	. . . . . . 7 ⊢ (𝐵 ∈ 𝑉 → ( ElDisj 𝐴 → ((𝑢 ∈ dom (◡ E ↾ 𝐴) ∧ 𝐵 ∈ [𝑢](◡ E ↾ 𝐴)) → [𝑢](◡ E ↾ 𝐴) = [𝐵] ≀ (◡ E ↾ 𝐴))))
4	3	imp 406	. . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ ElDisj 𝐴) → ((𝑢 ∈ dom (◡ E ↾ 𝐴) ∧ 𝐵 ∈ [𝑢](◡ E ↾ 𝐴)) → [𝑢](◡ E ↾ 𝐴) = [𝐵] ≀ (◡ E ↾ 𝐴)))
5	4	expdimp 452	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ ElDisj 𝐴) ∧ 𝑢 ∈ dom (◡ E ↾ 𝐴)) → (𝐵 ∈ [𝑢](◡ E ↾ 𝐴) → [𝑢](◡ E ↾ 𝐴) = [𝐵] ≀ (◡ E ↾ 𝐴)))
6		eccnvepres3 38462	. . . . . . . 8 ⊢ (𝑢 ∈ dom (◡ E ↾ 𝐴) → [𝑢](◡ E ↾ 𝐴) = 𝑢)
7	6	eleq2d 2821	. . . . . . 7 ⊢ (𝑢 ∈ dom (◡ E ↾ 𝐴) → (𝐵 ∈ [𝑢](◡ E ↾ 𝐴) ↔ 𝐵 ∈ 𝑢))
8	6	eqeq1d 2737	. . . . . . 7 ⊢ (𝑢 ∈ dom (◡ E ↾ 𝐴) → ([𝑢](◡ E ↾ 𝐴) = [𝐵] ≀ (◡ E ↾ 𝐴) ↔ 𝑢 = [𝐵] ≀ (◡ E ↾ 𝐴)))
9	7, 8	imbi12d 344	. . . . . 6 ⊢ (𝑢 ∈ dom (◡ E ↾ 𝐴) → ((𝐵 ∈ [𝑢](◡ E ↾ 𝐴) → [𝑢](◡ E ↾ 𝐴) = [𝐵] ≀ (◡ E ↾ 𝐴)) ↔ (𝐵 ∈ 𝑢 → 𝑢 = [𝐵] ≀ (◡ E ↾ 𝐴))))
10	9	adantl 481	. . . . 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 232	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ ElDisj 𝐴) ∧ 𝑢 ∈ dom (◡ E ↾ 𝐴)) → (𝐵 ∈ 𝑢 → 𝑢 = [𝐵] ≀ (◡ E ↾ 𝐴)))
12		df-coels 38672	. . . . . 6 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
13	12	eceq2i 8678	. . . . 5 ⊢ [𝐵] ∼ 𝐴 = [𝐵] ≀ (◡ E ↾ 𝐴)
14	13	eqeq2i 2748	. . . 4 ⊢ (𝑢 = [𝐵] ∼ 𝐴 ↔ 𝑢 = [𝐵] ≀ (◡ E ↾ 𝐴))
15	11, 14	imbitrrdi 252	. . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ ElDisj 𝐴) ∧ 𝑢 ∈ dom (◡ E ↾ 𝐴)) → (𝐵 ∈ 𝑢 → 𝑢 = [𝐵] ∼ 𝐴))
16	15	expimpd 453	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ ElDisj 𝐴) → ((𝑢 ∈ dom (◡ E ↾ 𝐴) ∧ 𝐵 ∈ 𝑢) → 𝑢 = [𝐵] ∼ 𝐴))
17	16	ex 412	1 ⊢ (𝐵 ∈ 𝑉 → ( ElDisj 𝐴 → ((𝑢 ∈ dom (◡ E ↾ 𝐴) ∧ 𝐵 ∈ 𝑢) → 𝑢 = [𝐵] ∼ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   E cep 5522  ^◡ccnv 5622  dom cdm 5623   ↾ cres 5625  [cec 8633   ≀ ccoss 38353   ∼ ccoels 38354   Disj wdisjALTV 38389   ElDisj weldisj 38391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3349  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-id 5518  df-eprel 5523  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ec 8637  df-coss 38671  df-coels 38672  df-cnvrefrel 38777  df-disjALTV 38960  df-eldisj 38962

This theorem is referenced by:  membpartlem19  39084