Theorem marypha2lem3 8343

Description: Lemma for marypha2 8345. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)

Hypothesis

Ref	Expression
marypha2lem.t	⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥))

Assertion

Ref	Expression
marypha2lem3	⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐺 ⊆ 𝑇 ↔ ∀𝑥 ∈ 𝐴 (𝐺‘𝑥) ∈ (𝐹‘𝑥)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺

Allowed substitution hint:   𝑇(𝑥)

Proof of Theorem marypha2lem3

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffn5 6241	. . . . . . 7 ⊢ (𝐺 Fn 𝐴 ↔ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)))
2	1	biimpi 206	. . . . . 6 ⊢ (𝐺 Fn 𝐴 → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)))
3	2	adantl 482	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)))
4		df-mpt 4730	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))}
5	3, 4	syl6eq 2672	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))})
6		marypha2lem.t	. . . . . 6 ⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥))
7	6	marypha2lem2 8342	. . . . 5 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
8	7	a1i 11	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))})
9	5, 8	sseq12d 3634	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐺 ⊆ 𝑇 ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}))
10		ssopab2b 5002	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))))
11	9, 10	syl6bb 276	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐺 ⊆ 𝑇 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)))))
12		19.21v 1868	. . . . 5 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → (𝑦 = (𝐺‘𝑥) → 𝑦 ∈ (𝐹‘𝑥))) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 = (𝐺‘𝑥) → 𝑦 ∈ (𝐹‘𝑥))))
13		imdistan 725	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → (𝑦 = (𝐺‘𝑥) → 𝑦 ∈ (𝐹‘𝑥))) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))))
14	13	albii 1747	. . . . 5 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → (𝑦 = (𝐺‘𝑥) → 𝑦 ∈ (𝐹‘𝑥))) ↔ ∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))))
15		fvex 6201	. . . . . . 7 ⊢ (𝐺‘𝑥) ∈ V
16		eleq1 2689	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑥) → (𝑦 ∈ (𝐹‘𝑥) ↔ (𝐺‘𝑥) ∈ (𝐹‘𝑥)))
17	15, 16	ceqsalv 3233	. . . . . 6 ⊢ (∀𝑦(𝑦 = (𝐺‘𝑥) → 𝑦 ∈ (𝐹‘𝑥)) ↔ (𝐺‘𝑥) ∈ (𝐹‘𝑥))
18	17	imbi2i 326	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → ∀𝑦(𝑦 = (𝐺‘𝑥) → 𝑦 ∈ (𝐹‘𝑥))) ↔ (𝑥 ∈ 𝐴 → (𝐺‘𝑥) ∈ (𝐹‘𝑥)))
19	12, 14, 18	3bitr3i 290	. . . 4 ⊢ (∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))) ↔ (𝑥 ∈ 𝐴 → (𝐺‘𝑥) ∈ (𝐹‘𝑥)))
20	19	albii 1747	. . 3 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝐺‘𝑥) ∈ (𝐹‘𝑥)))
21		df-ral 2917	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝐺‘𝑥) ∈ (𝐹‘𝑥) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝐺‘𝑥) ∈ (𝐹‘𝑥)))
22	20, 21	bitr4i 267	. 2 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))) ↔ ∀𝑥 ∈ 𝐴 (𝐺‘𝑥) ∈ (𝐹‘𝑥))
23	11, 22	syl6bb 276	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐺 ⊆ 𝑇 ↔ ∀𝑥 ∈ 𝐴 (𝐺‘𝑥) ∈ (𝐹‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ⊆ wss 3574  {csn 4177  ∪ ciun 4520  {copab 4712   ↦ cmpt 4729   × cxp 5112   Fn wfn 5883  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  marypha2  8345