Theorem eldisjlem19 38791

Description: Special case of disjlem19 38782 (together with membpartlem19 38792, this is former prtlem19 38859). (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 38688	. . . . . . . 8 ⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))
2		disjlem19 38782	. . . . . . . 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 38267	. . . . . . . 8 ⊢ (𝑢 ∈ dom (◡ E ↾ 𝐴) → [𝑢](◡ E ↾ 𝐴) = 𝑢)
7	6	eleq2d 2824	. . . . . . 7 ⊢ (𝑢 ∈ dom (◡ E ↾ 𝐴) → (𝐵 ∈ [𝑢](◡ E ↾ 𝐴) ↔ 𝐵 ∈ 𝑢))
8	6	eqeq1d 2736	. . . . . . 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 38393	. . . . . 6 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
13	12	eceq2i 8785	. . . . 5 ⊢ [𝐵] ∼ 𝐴 = [𝐵] ≀ (◡ E ↾ 𝐴)
14	13	eqeq2i 2747	. . . 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 1536   ∈ wcel 2105   E cep 5587  ^◡ccnv 5687  dom cdm 5688   ↾ cres 5690  [cec 8741   ≀ ccoss 38161   ∼ ccoels 38162   Disj wdisjALTV 38195   ElDisj weldisj 38197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-id 5582  df-eprel 5588  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ec 8745  df-coss 38392  df-coels 38393  df-cnvrefrel 38508  df-disjALTV 38686  df-eldisj 38688

This theorem is referenced by:  membpartlem19  38792