Theorem frege118 44258

Description: Simplified application of one direction of dffrege115 44255. 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 44198	. . 3 ⊢ (∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)) → [𝑌 / 𝑏](𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))
3		sbcimg 3790	. . . . 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 5156	. . . . . . 7 ⊢ (𝑌 ∈ 𝑉 → ([𝑌 / 𝑏]𝑏𝑅𝑋 ↔ ⦋𝑌 / 𝑏⦌𝑏𝑅𝑋))
6	1, 5	ax-mp 5	. . . . . 6 ⊢ ([𝑌 / 𝑏]𝑏𝑅𝑋 ↔ ⦋𝑌 / 𝑏⦌𝑏𝑅𝑋)
7		csbvarg 4387	. . . . . . . 8 ⊢ (𝑌 ∈ 𝑉 → ⦋𝑌 / 𝑏⦌𝑏 = 𝑌)
8	1, 7	ax-mp 5	. . . . . . 7 ⊢ ⦋𝑌 / 𝑏⦌𝑏 = 𝑌
9	8	breq1i 5106	. . . . . 6 ⊢ (⦋𝑌 / 𝑏⦌𝑏𝑅𝑋 ↔ 𝑌𝑅𝑋)
10	6, 9	bitri 275	. . . . 5 ⊢ ([𝑌 / 𝑏]𝑏𝑅𝑋 ↔ 𝑌𝑅𝑋)
11		sbcal 3801	. . . . . 6 ⊢ ([𝑌 / 𝑏]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋) ↔ ∀𝑎[𝑌 / 𝑏](𝑏𝑅𝑎 → 𝑎 = 𝑋))
12		sbcimg 3790	. . . . . . . . 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 5156	. . . . . . . . . . 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 5106	. . . . . . . . . 10 ⊢ (⦋𝑌 / 𝑏⦌𝑏𝑅𝑎 ↔ 𝑌𝑅𝑎)
17	15, 16	bitri 275	. . . . . . . . 9 ⊢ ([𝑌 / 𝑏]𝑏𝑅𝑎 ↔ 𝑌𝑅𝑎)
18		sbcg 3814	. . . . . . . . . 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 275	. . . . . . 7 ⊢ ([𝑌 / 𝑏](𝑏𝑅𝑎 → 𝑎 = 𝑋) ↔ (𝑌𝑅𝑎 → 𝑎 = 𝑋))
22	21	albii 1821	. . . . . 6 ⊢ (∀𝑎[𝑌 / 𝑏](𝑏𝑅𝑎 → 𝑎 = 𝑋) ↔ ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋))
23	11, 22	bitri 275	. . . . 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 275	. . 3 ⊢ ([𝑌 / 𝑏](𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)) ↔ (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋)))
26	2, 25	sylib 218	. 2 ⊢ (∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)) → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋)))
27		frege116.x	. . 3 ⊢ 𝑋 ∈ 𝑈
28	27	frege117 44257	. 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 206  ∀wal 1540   = wceq 1542   ∈ wcel 2114  [wsbc 3741  ⦋csb 3850   class class class wbr 5099  ^◡ccnv 5624  Fun wfun 6487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-frege1 44067  ax-frege2 44068  ax-frege8 44086  ax-frege52a 44134  ax-frege58b 44178

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-fun 6495

This theorem is referenced by:  frege119  44259