Theorem fnwe2lem1 41792

Description: Lemma for fnwe2 41795. 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 3147	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)})
3		fveq2 6892	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (𝐹‘𝑎) = (𝐹‘𝑥))
4	3	csbeq1d 3898	. . . . . 6 ⊢ (𝑎 = 𝑥 → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 = ⦋(𝐹‘𝑥) / 𝑧⦌𝑆)
5		fvex 6905	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
6		fnwe2.su	. . . . . . 7 ⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈)
7	5, 6	csbie 3930	. . . . . 6 ⊢ ⦋(𝐹‘𝑥) / 𝑧⦌𝑆 = 𝑈
8	4, 7	eqtrdi 2789	. . . . 5 ⊢ (𝑎 = 𝑥 → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 = 𝑈)
9	3	eqeq2d 2744	. . . . . 6 ⊢ (𝑎 = 𝑥 → ((𝐹‘𝑦) = (𝐹‘𝑎) ↔ (𝐹‘𝑦) = (𝐹‘𝑥)))
10	9	rabbidv 3441	. . . . 5 ⊢ (𝑎 = 𝑥 → {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)} = {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)})
11	8, 10	weeq12d 41782	. . . 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 3235	. . 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 233	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)})
14	13	r19.21bi 3249	1 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)})

Syntax hints:   → wi 4   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107  ∀wral 3062  {crab 3433  ⦋csb 3894   class class class wbr 5149  {copab 5211   We wwe 5631  ‘cfv 6544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2704  ax-nul 5307

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-iota 6496  df-fv 6552

This theorem is referenced by:  fnwe2lem2  41793  fnwe2lem3  41794