Theorem frege116 42812

Description: One direction of dffrege115 42811. Proposition 116 of [Frege1879] p. 77. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege116.x	⊢ 𝑋 ∈ 𝑈

Assertion

Ref	Expression
frege116	⊢ (Fun ◡◡𝑅 → ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))

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

Allowed substitution hints:   𝑈(𝑎,𝑏)

Proof of Theorem frege116

Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffrege115 42811	. 2 ⊢ (∀𝑐∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ Fun ◡◡𝑅)
2		frege116.x	. . . 4 ⊢ 𝑋 ∈ 𝑈
3	2	frege68c 42764	. . 3 ⊢ ((∀𝑐∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ Fun ◡◡𝑅) → (Fun ◡◡𝑅 → [𝑋 / 𝑐]∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐))))
4		sbcal 3841	. . . 4 ⊢ ([𝑋 / 𝑐]∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑏[𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)))
5		sbcimg 3828	. . . . . . 7 ⊢ (𝑋 ∈ 𝑈 → ([𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ([𝑋 / 𝑐]𝑏𝑅𝑐 → [𝑋 / 𝑐]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐))))
6	2, 5	ax-mp 5	. . . . . 6 ⊢ ([𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ([𝑋 / 𝑐]𝑏𝑅𝑐 → [𝑋 / 𝑐]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)))
7		sbcbr2g 5206	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑈 → ([𝑋 / 𝑐]𝑏𝑅𝑐 ↔ 𝑏𝑅⦋𝑋 / 𝑐⦌𝑐))
8	2, 7	ax-mp 5	. . . . . . . 8 ⊢ ([𝑋 / 𝑐]𝑏𝑅𝑐 ↔ 𝑏𝑅⦋𝑋 / 𝑐⦌𝑐)
9		csbvarg 4431	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑈 → ⦋𝑋 / 𝑐⦌𝑐 = 𝑋)
10	2, 9	ax-mp 5	. . . . . . . . 9 ⊢ ⦋𝑋 / 𝑐⦌𝑐 = 𝑋
11	10	breq2i 5156	. . . . . . . 8 ⊢ (𝑏𝑅⦋𝑋 / 𝑐⦌𝑐 ↔ 𝑏𝑅𝑋)
12	8, 11	bitri 274	. . . . . . 7 ⊢ ([𝑋 / 𝑐]𝑏𝑅𝑐 ↔ 𝑏𝑅𝑋)
13		sbcal 3841	. . . . . . . 8 ⊢ ([𝑋 / 𝑐]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐) ↔ ∀𝑎[𝑋 / 𝑐](𝑏𝑅𝑎 → 𝑎 = 𝑐))
14		sbcimg 3828	. . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝑈 → ([𝑋 / 𝑐](𝑏𝑅𝑎 → 𝑎 = 𝑐) ↔ ([𝑋 / 𝑐]𝑏𝑅𝑎 → [𝑋 / 𝑐]𝑎 = 𝑐)))
15	2, 14	ax-mp 5	. . . . . . . . . 10 ⊢ ([𝑋 / 𝑐](𝑏𝑅𝑎 → 𝑎 = 𝑐) ↔ ([𝑋 / 𝑐]𝑏𝑅𝑎 → [𝑋 / 𝑐]𝑎 = 𝑐))
16		sbcg 3856	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝑈 → ([𝑋 / 𝑐]𝑏𝑅𝑎 ↔ 𝑏𝑅𝑎))
17	2, 16	ax-mp 5	. . . . . . . . . . 11 ⊢ ([𝑋 / 𝑐]𝑏𝑅𝑎 ↔ 𝑏𝑅𝑎)
18		sbceq2g 4416	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝑈 → ([𝑋 / 𝑐]𝑎 = 𝑐 ↔ 𝑎 = ⦋𝑋 / 𝑐⦌𝑐))
19	2, 18	ax-mp 5	. . . . . . . . . . . 12 ⊢ ([𝑋 / 𝑐]𝑎 = 𝑐 ↔ 𝑎 = ⦋𝑋 / 𝑐⦌𝑐)
20	10	eqeq2i 2745	. . . . . . . . . . . 12 ⊢ (𝑎 = ⦋𝑋 / 𝑐⦌𝑐 ↔ 𝑎 = 𝑋)
21	19, 20	bitri 274	. . . . . . . . . . 11 ⊢ ([𝑋 / 𝑐]𝑎 = 𝑐 ↔ 𝑎 = 𝑋)
22	17, 21	imbi12i 350	. . . . . . . . . 10 ⊢ (([𝑋 / 𝑐]𝑏𝑅𝑎 → [𝑋 / 𝑐]𝑎 = 𝑐) ↔ (𝑏𝑅𝑎 → 𝑎 = 𝑋))
23	15, 22	bitri 274	. . . . . . . . 9 ⊢ ([𝑋 / 𝑐](𝑏𝑅𝑎 → 𝑎 = 𝑐) ↔ (𝑏𝑅𝑎 → 𝑎 = 𝑋))
24	23	albii 1821	. . . . . . . 8 ⊢ (∀𝑎[𝑋 / 𝑐](𝑏𝑅𝑎 → 𝑎 = 𝑐) ↔ ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋))
25	13, 24	bitri 274	. . . . . . 7 ⊢ ([𝑋 / 𝑐]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐) ↔ ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋))
26	12, 25	imbi12i 350	. . . . . 6 ⊢ (([𝑋 / 𝑐]𝑏𝑅𝑐 → [𝑋 / 𝑐]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ (𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))
27	6, 26	bitri 274	. . . . 5 ⊢ ([𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ (𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))
28	27	albii 1821	. . . 4 ⊢ (∀𝑏[𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))
29	4, 28	bitri 274	. . 3 ⊢ ([𝑋 / 𝑐]∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))
30	3, 29	imbitrdi 250	. 2 ⊢ ((∀𝑐∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ Fun ◡◡𝑅) → (Fun ◡◡𝑅 → ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋))))
31	1, 30	ax-mp 5	1 ⊢ (Fun ◡◡𝑅 → ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1539   = wceq 1541   ∈ wcel 2106  [wsbc 3777  ⦋csb 3893   class class class wbr 5148  ^◡ccnv 5675  Fun wfun 6537

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-frege1 42623  ax-frege2 42624  ax-frege8 42642  ax-frege52a 42690  ax-frege58b 42734

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-fun 6545

This theorem is referenced by:  frege117  42813