Theorem marypha2lem3 9321

Description: Lemma for marypha2 9323. 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 6880	. . . . . . 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 5171	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))}
5	3, 4	eqtrdi 2782	. . . 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 9320	. . . . 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 3963	. . 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 5485	. . 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 1940	. . . . 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 1820	. . . . 5 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → (𝑦 = (𝐺‘𝑥) → 𝑦 ∈ (𝐹‘𝑥))) ↔ ∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))))
15		fvex 6835	. . . . . . 7 ⊢ (𝐺‘𝑥) ∈ V
16		eleq1 2819	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑥) → (𝑦 ∈ (𝐹‘𝑥) ↔ (𝐺‘𝑥) ∈ (𝐹‘𝑥)))
17	15, 16	ceqsalv 3476	. . . . . 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 1820	. . 3 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝐺‘𝑥) ∈ (𝐹‘𝑥)))
21		df-ral 3048	. . 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 1539   = wceq 1541   ∈ wcel 2111  ∀wral 3047   ⊆ wss 3897  {csn 4573  ∪ ciun 4939  {copab 5151   ↦ cmpt 5170   × cxp 5612   Fn wfn 6476  ‘cfv 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6437  df-fun 6483  df-fn 6484  df-fv 6489

This theorem is referenced by:  marypha2  9323