Theorem frege116 44438

Description: One direction of dffrege115 44437. 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 44437	. 2 ⊢ (∀𝑐∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ Fun ◡◡𝑅)
2		frege116.x	. . . 4 ⊢ 𝑋 ∈ 𝑈
3	2	frege68c 44390	. . 3 ⊢ ((∀𝑐∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ Fun ◡◡𝑅) → (Fun ◡◡𝑅 → [𝑋 / 𝑐]∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐))))
4		sbcal 3784	. . . 4 ⊢ ([𝑋 / 𝑐]∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑏[𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)))
5		sbcimg 3773	. . . . . . 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 5133	. . . . . . . . 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 4365	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑈 → ⦋𝑋 / 𝑐⦌𝑐 = 𝑋)
10	2, 9	ax-mp 5	. . . . . . . . 9 ⊢ ⦋𝑋 / 𝑐⦌𝑐 = 𝑋
11	10	breq2i 5083	. . . . . . . 8 ⊢ (𝑏𝑅⦋𝑋 / 𝑐⦌𝑐 ↔ 𝑏𝑅𝑋)
12	8, 11	bitri 277	. . . . . . 7 ⊢ ([𝑋 / 𝑐]𝑏𝑅𝑐 ↔ 𝑏𝑅𝑋)
13		sbcal 3784	. . . . . . . 8 ⊢ ([𝑋 / 𝑐]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐) ↔ ∀𝑎[𝑋 / 𝑐](𝑏𝑅𝑎 → 𝑎 = 𝑐))
14		sbcimg 3773	. . . . . . . . . . 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 3797	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝑈 → ([𝑋 / 𝑐]𝑏𝑅𝑎 ↔ 𝑏𝑅𝑎))
17	2, 16	ax-mp 5	. . . . . . . . . . 11 ⊢ ([𝑋 / 𝑐]𝑏𝑅𝑎 ↔ 𝑏𝑅𝑎)
18		sbceq2g 4350	. . . . . . . . . . . . 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 2754	. . . . . . . . . . . 12 ⊢ (𝑎 = ⦋𝑋 / 𝑐⦌𝑐 ↔ 𝑎 = 𝑋)
21	19, 20	bitri 277	. . . . . . . . . . 11 ⊢ ([𝑋 / 𝑐]𝑎 = 𝑐 ↔ 𝑎 = 𝑋)
22	17, 21	imbi12i 352	. . . . . . . . . 10 ⊢ (([𝑋 / 𝑐]𝑏𝑅𝑎 → [𝑋 / 𝑐]𝑎 = 𝑐) ↔ (𝑏𝑅𝑎 → 𝑎 = 𝑋))
23	15, 22	bitri 277	. . . . . . . . 9 ⊢ ([𝑋 / 𝑐](𝑏𝑅𝑎 → 𝑎 = 𝑐) ↔ (𝑏𝑅𝑎 → 𝑎 = 𝑋))
24	23	albii 1827	. . . . . . . 8 ⊢ (∀𝑎[𝑋 / 𝑐](𝑏𝑅𝑎 → 𝑎 = 𝑐) ↔ ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋))
25	13, 24	bitri 277	. . . . . . 7 ⊢ ([𝑋 / 𝑐]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐) ↔ ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋))
26	12, 25	imbi12i 352	. . . . . 6 ⊢ (([𝑋 / 𝑐]𝑏𝑅𝑐 → [𝑋 / 𝑐]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ (𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))
27	6, 26	bitri 277	. . . . 5 ⊢ ([𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ (𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))
28	27	albii 1827	. . . 4 ⊢ (∀𝑏[𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))
29	4, 28	bitri 277	. . 3 ⊢ ([𝑋 / 𝑐]∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))
30	3, 29	imbitrdi 253	. 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 208  ∀wal 1546   = wceq 1548   ∈ wcel 2121  [wsbc 3725  ⦋csb 3833   class class class wbr 5075  ^◡ccnv 5620  Fun wfun 6483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-pr 5365  ax-frege1 44249  ax-frege2 44250  ax-frege8 44268  ax-frege52a 44316  ax-frege58b 44360

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-ifp 1070  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-fun 6491

This theorem is referenced by:  frege117  44439