Theorem marypha2lem3 9475

Description: Lemma for marypha2 9477. 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 6967	. . . . . . 7 ⊢ (𝐺 Fn 𝐴 ↔ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)))
2	1	biimpi 216	. . . . . 6 ⊢ (𝐺 Fn 𝐴 → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)))
3	2	adantl 481	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)))
4		df-mpt 5232	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))}
5	3, 4	eqtrdi 2791	. . . 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 9474	. . . . 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 4029	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐺 ⊆ 𝑇 ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}))
10		ssopab2bw 5557	. . 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	bitrdi 287	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐺 ⊆ 𝑇 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)))))
12		19.21v 1937	. . . . 5 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → (𝑦 = (𝐺‘𝑥) → 𝑦 ∈ (𝐹‘𝑥))) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 = (𝐺‘𝑥) → 𝑦 ∈ (𝐹‘𝑥))))
13		imdistan 567	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → (𝑦 = (𝐺‘𝑥) → 𝑦 ∈ (𝐹‘𝑥))) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))))
14	13	albii 1816	. . . . 5 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → (𝑦 = (𝐺‘𝑥) → 𝑦 ∈ (𝐹‘𝑥))) ↔ ∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))))
15		fvex 6920	. . . . . . 7 ⊢ (𝐺‘𝑥) ∈ V
16		eleq1 2827	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑥) → (𝑦 ∈ (𝐹‘𝑥) ↔ (𝐺‘𝑥) ∈ (𝐹‘𝑥)))
17	15, 16	ceqsalv 3519	. . . . . 6 ⊢ (∀𝑦(𝑦 = (𝐺‘𝑥) → 𝑦 ∈ (𝐹‘𝑥)) ↔ (𝐺‘𝑥) ∈ (𝐹‘𝑥))
18	17	imbi2i 336	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → ∀𝑦(𝑦 = (𝐺‘𝑥) → 𝑦 ∈ (𝐹‘𝑥))) ↔ (𝑥 ∈ 𝐴 → (𝐺‘𝑥) ∈ (𝐹‘𝑥)))
19	12, 14, 18	3bitr3i 301	. . . 4 ⊢ (∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))) ↔ (𝑥 ∈ 𝐴 → (𝐺‘𝑥) ∈ (𝐹‘𝑥)))
20	19	albii 1816	. . 3 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝐺‘𝑥) ∈ (𝐹‘𝑥)))
21		df-ral 3060	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝐺‘𝑥) ∈ (𝐹‘𝑥) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝐺‘𝑥) ∈ (𝐹‘𝑥)))
22	20, 21	bitr4i 278	. 2 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))) ↔ ∀𝑥 ∈ 𝐴 (𝐺‘𝑥) ∈ (𝐹‘𝑥))
23	11, 22	bitrdi 287	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐺 ⊆ 𝑇 ↔ ∀𝑥 ∈ 𝐴 (𝐺‘𝑥) ∈ (𝐹‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535   = wceq 1537   ∈ wcel 2106  ∀wral 3059   ⊆ wss 3963  {csn 4631  ∪ ciun 4996  {copab 5210   ↦ cmpt 5231   × cxp 5687   Fn wfn 6558  ‘cfv 6563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571

This theorem is referenced by:  marypha2  9477