Theorem frege118 41478

Description: Simplified application of one direction of dffrege115 41475. Proposition 118 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
frege116.x	⊢ 𝑋 ∈ 𝑈
frege118.y	⊢ 𝑌 ∈ 𝑉

Assertion

Ref	Expression
frege118	⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋)))

Distinct variable groups:   𝑅,𝑎   𝑋,𝑎   𝑌,𝑎

Allowed substitution hints:   𝑈(𝑎)   𝑉(𝑎)

Proof of Theorem frege118

Dummy variable 𝑏 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frege118.y	. . . 4 ⊢ 𝑌 ∈ 𝑉
2	1	frege58c 41418	. . 3 ⊢ (∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)) → [𝑌 / 𝑏](𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))
3		sbcimg 3762	. . . . 5 ⊢ (𝑌 ∈ 𝑉 → ([𝑌 / 𝑏](𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)) ↔ ([𝑌 / 𝑏]𝑏𝑅𝑋 → [𝑌 / 𝑏]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋))))
4	1, 3	ax-mp 5	. . . 4 ⊢ ([𝑌 / 𝑏](𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)) ↔ ([𝑌 / 𝑏]𝑏𝑅𝑋 → [𝑌 / 𝑏]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))
5		sbcbr1g 5127	. . . . . . 7 ⊢ (𝑌 ∈ 𝑉 → ([𝑌 / 𝑏]𝑏𝑅𝑋 ↔ ⦋𝑌 / 𝑏⦌𝑏𝑅𝑋))
6	1, 5	ax-mp 5	. . . . . 6 ⊢ ([𝑌 / 𝑏]𝑏𝑅𝑋 ↔ ⦋𝑌 / 𝑏⦌𝑏𝑅𝑋)
7		csbvarg 4362	. . . . . . . 8 ⊢ (𝑌 ∈ 𝑉 → ⦋𝑌 / 𝑏⦌𝑏 = 𝑌)
8	1, 7	ax-mp 5	. . . . . . 7 ⊢ ⦋𝑌 / 𝑏⦌𝑏 = 𝑌
9	8	breq1i 5077	. . . . . 6 ⊢ (⦋𝑌 / 𝑏⦌𝑏𝑅𝑋 ↔ 𝑌𝑅𝑋)
10	6, 9	bitri 274	. . . . 5 ⊢ ([𝑌 / 𝑏]𝑏𝑅𝑋 ↔ 𝑌𝑅𝑋)
11		sbcal 3776	. . . . . 6 ⊢ ([𝑌 / 𝑏]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋) ↔ ∀𝑎[𝑌 / 𝑏](𝑏𝑅𝑎 → 𝑎 = 𝑋))
12		sbcimg 3762	. . . . . . . . 9 ⊢ (𝑌 ∈ 𝑉 → ([𝑌 / 𝑏](𝑏𝑅𝑎 → 𝑎 = 𝑋) ↔ ([𝑌 / 𝑏]𝑏𝑅𝑎 → [𝑌 / 𝑏]𝑎 = 𝑋)))
13	1, 12	ax-mp 5	. . . . . . . 8 ⊢ ([𝑌 / 𝑏](𝑏𝑅𝑎 → 𝑎 = 𝑋) ↔ ([𝑌 / 𝑏]𝑏𝑅𝑎 → [𝑌 / 𝑏]𝑎 = 𝑋))
14		sbcbr1g 5127	. . . . . . . . . . 11 ⊢ (𝑌 ∈ 𝑉 → ([𝑌 / 𝑏]𝑏𝑅𝑎 ↔ ⦋𝑌 / 𝑏⦌𝑏𝑅𝑎))
15	1, 14	ax-mp 5	. . . . . . . . . 10 ⊢ ([𝑌 / 𝑏]𝑏𝑅𝑎 ↔ ⦋𝑌 / 𝑏⦌𝑏𝑅𝑎)
16	8	breq1i 5077	. . . . . . . . . 10 ⊢ (⦋𝑌 / 𝑏⦌𝑏𝑅𝑎 ↔ 𝑌𝑅𝑎)
17	15, 16	bitri 274	. . . . . . . . 9 ⊢ ([𝑌 / 𝑏]𝑏𝑅𝑎 ↔ 𝑌𝑅𝑎)
18		sbcg 3791	. . . . . . . . . 10 ⊢ (𝑌 ∈ 𝑉 → ([𝑌 / 𝑏]𝑎 = 𝑋 ↔ 𝑎 = 𝑋))
19	1, 18	ax-mp 5	. . . . . . . . 9 ⊢ ([𝑌 / 𝑏]𝑎 = 𝑋 ↔ 𝑎 = 𝑋)
20	17, 19	imbi12i 350	. . . . . . . 8 ⊢ (([𝑌 / 𝑏]𝑏𝑅𝑎 → [𝑌 / 𝑏]𝑎 = 𝑋) ↔ (𝑌𝑅𝑎 → 𝑎 = 𝑋))
21	13, 20	bitri 274	. . . . . . 7 ⊢ ([𝑌 / 𝑏](𝑏𝑅𝑎 → 𝑎 = 𝑋) ↔ (𝑌𝑅𝑎 → 𝑎 = 𝑋))
22	21	albii 1823	. . . . . 6 ⊢ (∀𝑎[𝑌 / 𝑏](𝑏𝑅𝑎 → 𝑎 = 𝑋) ↔ ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋))
23	11, 22	bitri 274	. . . . 5 ⊢ ([𝑌 / 𝑏]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋) ↔ ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋))
24	10, 23	imbi12i 350	. . . 4 ⊢ (([𝑌 / 𝑏]𝑏𝑅𝑋 → [𝑌 / 𝑏]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)) ↔ (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋)))
25	4, 24	bitri 274	. . 3 ⊢ ([𝑌 / 𝑏](𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)) ↔ (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋)))
26	2, 25	sylib 217	. 2 ⊢ (∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)) → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋)))
27		frege116.x	. . 3 ⊢ 𝑋 ∈ 𝑈
28	27	frege117 41477	. 2 ⊢ ((∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)) → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋))) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋))))
29	26, 28	ax-mp 5	1 ⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋)))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537   = wceq 1539   ∈ wcel 2108  [wsbc 3711  ⦋csb 3828   class class class wbr 5070  ^◡ccnv 5579  Fun wfun 6412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-frege1 41287  ax-frege2 41288  ax-frege8 41306  ax-frege52a 41354  ax-frege58b 41398

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ifp 1060  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-fun 6420

This theorem is referenced by:  frege119  41479