Theorem fnwe2lem1 43032

Description: Lemma for fnwe2 43035. Substitution in well-ordering hypothesis. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Hypotheses

Ref	Expression
fnwe2.su	⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈)
fnwe2.t	⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦))}
fnwe2.s	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)})

Assertion

Ref	Expression
fnwe2lem1	⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)})

Distinct variable groups:   𝑦,𝑈,𝑧,𝑎   𝑥,𝑆,𝑦,𝑎   𝑥,𝑅,𝑦,𝑎   𝜑,𝑥,𝑦,𝑧   𝑥,𝐴,𝑦,𝑧,𝑎   𝑥,𝐹,𝑦,𝑧,𝑎   𝑇,𝑎

Allowed substitution hints:   𝜑(𝑎)   𝑅(𝑧)   𝑆(𝑧)   𝑇(𝑥,𝑦,𝑧)   𝑈(𝑥)

Proof of Theorem fnwe2lem1

Step	Hyp	Ref	Expression
1		fnwe2.s	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)})
2	1	ralrimiva 3126	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)})
3		fveq2 6860	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (𝐹‘𝑎) = (𝐹‘𝑥))
4	3	csbeq1d 3868	. . . . . 6 ⊢ (𝑎 = 𝑥 → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 = ⦋(𝐹‘𝑥) / 𝑧⦌𝑆)
5		fvex 6873	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
6		fnwe2.su	. . . . . . 7 ⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈)
7	5, 6	csbie 3899	. . . . . 6 ⊢ ⦋(𝐹‘𝑥) / 𝑧⦌𝑆 = 𝑈
8	4, 7	eqtrdi 2781	. . . . 5 ⊢ (𝑎 = 𝑥 → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 = 𝑈)
9	3	eqeq2d 2741	. . . . . 6 ⊢ (𝑎 = 𝑥 → ((𝐹‘𝑦) = (𝐹‘𝑎) ↔ (𝐹‘𝑦) = (𝐹‘𝑥)))
10	9	rabbidv 3416	. . . . 5 ⊢ (𝑎 = 𝑥 → {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)} = {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)})
11	8, 10	weeq12d 5629	. . . 4 ⊢ (𝑎 = 𝑥 → (⦋(𝐹‘𝑎) / 𝑧⦌𝑆 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)} ↔ 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)}))
12	11	cbvralvw 3216	. . 3 ⊢ (∀𝑎 ∈ 𝐴 ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)} ↔ ∀𝑥 ∈ 𝐴 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)})
13	2, 12	sylibr 234	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)})
14	13	r19.21bi 3230	1 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)})

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∀wral 3045  {crab 3408  ⦋csb 3864   class class class wbr 5109  {copab 5171   We wwe 5592  ‘cfv 6513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-12 2178  ax-ext 2702  ax-nul 5263

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-iota 6466  df-fv 6521

This theorem is referenced by:  fnwe2lem2  43033  fnwe2lem3  43034