Theorem frege120 44432

Description: Simplified application of one direction of dffrege115 44427. Proposition 120 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
frege120.a	⊢ 𝐴 ∈ 𝑊

Assertion

Ref	Expression
frege120	⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝐴 = 𝑋)))

Proof of Theorem frege120

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frege120.a	. . . 4 ⊢ 𝐴 ∈ 𝑊
2	1	frege58c 44370	. . 3 ⊢ (∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋) → [𝐴 / 𝑎](𝑌𝑅𝑎 → 𝑎 = 𝑋))
3		sbcim1 3783	. . . 4 ⊢ ([𝐴 / 𝑎](𝑌𝑅𝑎 → 𝑎 = 𝑋) → ([𝐴 / 𝑎]𝑌𝑅𝑎 → [𝐴 / 𝑎]𝑎 = 𝑋))
4		sbcbr2g 5144	. . . . . 6 ⊢ (𝐴 ∈ 𝑊 → ([𝐴 / 𝑎]𝑌𝑅𝑎 ↔ 𝑌𝑅⦋𝐴 / 𝑎⦌𝑎))
5	1, 4	ax-mp 5	. . . . 5 ⊢ ([𝐴 / 𝑎]𝑌𝑅𝑎 ↔ 𝑌𝑅⦋𝐴 / 𝑎⦌𝑎)
6		csbvarg 4375	. . . . . . 7 ⊢ (𝐴 ∈ 𝑊 → ⦋𝐴 / 𝑎⦌𝑎 = 𝐴)
7	1, 6	ax-mp 5	. . . . . 6 ⊢ ⦋𝐴 / 𝑎⦌𝑎 = 𝐴
8	7	breq2i 5094	. . . . 5 ⊢ (𝑌𝑅⦋𝐴 / 𝑎⦌𝑎 ↔ 𝑌𝑅𝐴)
9	5, 8	bitri 275	. . . 4 ⊢ ([𝐴 / 𝑎]𝑌𝑅𝑎 ↔ 𝑌𝑅𝐴)
10		sbceq1g 4358	. . . . . 6 ⊢ (𝐴 ∈ 𝑊 → ([𝐴 / 𝑎]𝑎 = 𝑋 ↔ ⦋𝐴 / 𝑎⦌𝑎 = 𝑋))
11	1, 10	ax-mp 5	. . . . 5 ⊢ ([𝐴 / 𝑎]𝑎 = 𝑋 ↔ ⦋𝐴 / 𝑎⦌𝑎 = 𝑋)
12	7	eqeq1i 2742	. . . . 5 ⊢ (⦋𝐴 / 𝑎⦌𝑎 = 𝑋 ↔ 𝐴 = 𝑋)
13	11, 12	bitri 275	. . . 4 ⊢ ([𝐴 / 𝑎]𝑎 = 𝑋 ↔ 𝐴 = 𝑋)
14	3, 9, 13	3imtr3g 295	. . 3 ⊢ ([𝐴 / 𝑎](𝑌𝑅𝑎 → 𝑎 = 𝑋) → (𝑌𝑅𝐴 → 𝐴 = 𝑋))
15	2, 14	syl 17	. 2 ⊢ (∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋) → (𝑌𝑅𝐴 → 𝐴 = 𝑋))
16		frege116.x	. . 3 ⊢ 𝑋 ∈ 𝑈
17		frege118.y	. . 3 ⊢ 𝑌 ∈ 𝑉
18	16, 17	frege119 44431	. 2 ⊢ ((∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋) → (𝑌𝑅𝐴 → 𝐴 = 𝑋)) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝐴 = 𝑋))))
19	15, 18	ax-mp 5	1 ⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝐴 = 𝑋)))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1540   = wceq 1542   ∈ wcel 2114  [wsbc 3729  ⦋csb 3838   class class class wbr 5086  ^◡ccnv 5625  Fun wfun 6488

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 5232  ax-pr 5372  ax-frege1 44239  ax-frege2 44240  ax-frege8 44258  ax-frege52a 44306  ax-frege58b 44350

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 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-fun 6496

This theorem is referenced by:  frege121  44433